Engineering the Coupling of Superconducting Qubits

Engineering the coupling of superconducting qubits

Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

M. Sc. Alessandro Ciani aus Penne, Italien

Berichter: Prof. Dr. David DiVincenzo Prof. Dr. Fabian Hassler

Tag der mündlichen Prüfung: 12 April 2019

Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek verfügbar. ii

«Considerate la vostra semenza: fatti non foste a viver come bruti, ma per seguir virtute e conoscenza.»

Dante Alighieri, "La Divina Commedia", Inferno, Canto XXVI, vv. 118-120. iii

Abstract

The way to build a scalable and reliable quantum computer that truly exploits the quantum power faces several challenges. Among the various proposals for building a quantum computer, superconducting qubits have rapidly progressed and hold good promises in the near-term future. In particular, the possibility to design the required interactions is one of the most appealing features of this kind of architecture. This thesis deals with some detailed aspects of this problem focusing on architectures based on superconducting transmon-like qubits. After reviewing the basic tools needed for the study of superconducting circuits and the main kinds of superconducting qubits, we move to the analyisis of a possible scheme for realizing direct parity measurement. Parity measurements, or in general stabilizer measurements, are fundamental tools for realizing quantum error correcting codes, that are believed to be fundamental for dealing with the problem of decoherence that affects any physical implementation of a quantum computer. While these measurements are usually done indirectly with the help of ancilla qubits, the scheme that we analyze performs the measurement directly, and requires the engineering of a precise matching condition. We show how sufﬁcient freedom in the design of the interactions can be achieved with tunable coupling qubits, which are a variant of transmon qubits. In the second part of the thesis, we study instead an alternative model for quantum computation and a possible realization with transmon qubits. The model performs a quantum computation with a time-independent Hamiltonian and it is closely connected to one of the original proposals for a quantum computer due to Feynman. After explaining the basic ideas of the model, we also discuss a new version with a modiﬁed dynamics compared to previous proposals, which maps exactly to Feyn- man’s original idea, and also how to realize a Toffoli gate in the model. We then move to the analysis of an implementation with transmon qubits. We show how it is possible to engineer the Hamiltonian that performs the desired computation in a completely passive way, and with the desired range of parameters, limiting spurious, unwanted terms.

Zusammenfassung

Die Realisierung eines skalierbaren und zuverlässigen Quantencomputers, der tat- sächlich die Quanten-Vorteile ausnutzt, steht vor mehreren Herausforderungen. Un- ter den verschiedenen Vorschlägen einer Implementierung eines Quantencomputers haben sich supraleitende Qubits rasch verbessert und sind sehr vielversprechend in absehbarer Zukunft. Insbesondere die Möglichkeit, die benötigten Wechselwirkun- gen einzustellen, ist eine der attraktivsten Eigenschaften dieser Architektur. Diese Dissertation handelt von einigen detaillierten Aspekten dieses Problems, wobei Ar- chitekturen, welche auf supraleitende Qubits ähnlich dem Transmon basieren, im Vordergrund stehen. Nach der Besprechung grundlegender Werkzeuge, die für die Untersuchung supraleitender Schaltkreise benötigt werden, sowie der wichtigsten supraleitenden Qubits analysieren wir einen möglichen Entwurf für die Realisierung einer direkten Paritätsmessung. Die Messungen der Parität, bzw. i.A. sogenannte „stabilizer“ Mes- sungen, sind fundamentale Werkzeuge für die Realisierung einer Quantenfehlerkor- rektur. Diese ist fundamental, um das Problem der Dekohärenz, die jede physikali- sche Implementierung eines Quantencomputers beeinﬂusst, zu umgehen. Während diese Messungen normalerweise indirekt mit Hilfs-Qubits ausgeführt werden, analysieren wir ein Schema, das diese Messungen direkt ausführt, jedoch die Anpas- sung einer genauen Bedingung voraussetzt. Wir zeigen, wie ausreichend Freiheiten in der Gestaltung der Wechselwirkung mit „tunable coupling qubits“, die eine Vari- ante des Transmons sind, erreicht werden können. Im zweiten Teil dieser Dissertation betrachten wir ein alternatives Modell eines Quantencomputers sowie eine mögliche Realisierung mit Transmons. Dieses Modell realisiert einen Quantencomputer mit einem zeitunabhängigen Hamilton-Operator und ist nah verwandt mit dem ursprünglichen Modell eines Quantencomputers, das von Feynman vorgeschlagen wurde. Nach der Einführung der Grundideen dieses Modells analysieren wir eine neue Version mit modiﬁzierter Dynamik, welche exakt mit Feynmans ursprünglicher Idee übereinstimmt. Außerdem wird gezeigt, wie das Toffoli-Gatter in diesem Modell realisiert wird. Im Anschluss folgt eine Analyse der Implementierung mit Transmons. Wir zeigen, wie es möglich ist einen Hamilton- Operator zu konstruieren, der die gewünschte Rechnung in einer komplett passiven Arten und Weise sowie mit der gewünschten Bandbreite der Parameter ausführt, wobei störende, unerwünschte Terme limitiert werden.

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Acknowledgements

During these years I’ve always thought about the moment in which I would have written the acknowledgements of my thesis, and finally on a Sunday morning here I am. I would like to begin by thanking my supervisor David DiVincenzo, who almost four years ago believed in a crazy italian engineer who wanted to move to quantum physics. Thank you for guiding me during these years and patiently teaching me how to be a physicist through all the daily difficulties. I deeply admire your passion and curiosity towards discovering new things, and I hope that this passion will also always be with me for the rest of my life. I also thank Barbara Terhal for giving me the opportunity to work with her and for the contagious entuhsiasm that she puts in her job. I really enjoyed working with you and I thank you for all I learned, and I continue to learn from you, and I mean not only about phyiscs. My thanks go also to Fabian Hassler who taught me the importance of developing a physical intuition in any problem we treat. Thanks also for reviewing this thesis and for always being available for discussions when I was randomly stepping into your office. I deeply thank all my friends of the IQI that I met in these years. First of all, I would like to thank my legendary office mate Stefano Bosco, with whom I shared the office for almost all the period of my PhD. I loved our fights about physics, in particular about Bloch’s theorem in presence of magnetic field, and I will miss our daily office life. Also thank you for fostering my knowledge of italian trash music. I also thank my first office mate Firat Solgun for welcoming me when I arrived here. I never told you that the first night we went out together I really disliked the turkish food we had, and it is finally time for the truth. I would also like to thank all the people who shared the office with me even only for a brief period: Jonathan Conrad (also for helping me with my exam), Joris Dolderer, Sander Konijnenberg, Fatemeh Hajiloo, Eva Fluck and Rijul Sachdeva. Thank you Manuel Rispler and Susanne Richer for being my patient german teachers during these years. I thank Manuel also for helping me through the crazy paperwork of german burocracy, and Susanne for always reminding me how good I am in quitting smoking. I thank also Cica Gustiani for the "amazing" indonesian cigarettes she brought back for me from Indonesia. I thank Nikolas Breuckmann for always sharing provoking pictures of weird pizzas with me. I thank Ben Criger for explaining to me many things about the canadian culture, and in particular who Wayne Gretzki is. I thank also Ben and Jakob Stubenrauch for our works together. Special thanks go also to Martin Rymarz who helped me translating the abstract of this thesis, and for all the help he provided in the last period. I deeply appreciate the fact that you are believing in the "liketattico" project and developing the app in this very moment. viii

I continue by thanking Daniel Otten for giving me the opportunity to become the legendary "spanish guy" of quantum optics, and for sharing with me this last period of thesis submission. I also thank Federica Haupt, in particular for giving to me as a present a mixer , that exploded in my hands few days ago, and for really nice dinners, and also Ananda Roy for listening to me when I said that the Guns n’ Roses are (by far) the best band of all times, and for explaining to us the correct and non-trivial pronounciation of his name (no, it is not Amanda!). I thank Fabio Pedrocchi for all the time we spent at the gym together without results. I thank Benedikt Placke for teaching me that golf is indeed a sport, and Lucia Gonzalez Rosado, for telling me that it is possible to survive one year of erasmus cooking only with the microwave oven. I thank Alexander Ziesen for still believing in the fact that one day I will come back to play football with the IQI. My special thanks go also to Lisa Arndt for her always positive mood, and Uta Meyer, for her resilience to the above-mentioned italian trash music. I also spent a really nice time with Jascha Ulrich and Amin Hosseinkhani at the conference of the condensed matter division of the European Physical society in Groningen, and with Roman Riwar and Kenneth Goodenough at the APS meeting in New Orleans, and I thank you for those nice moments. I am also grateful to Gianluigi Catelani for inviting me to the conference in Groningen and to Shabir Barzanjeh for the invitation to the conference "Frontiers in circuit QED" in Vienna. I would also like to thank Andrew Goldsborough for his extremely accurate political predictions of the last years. I thank Maarten Wegewjis for all the times I stole coffee from him, and Norbert Schuch for the answers he gave me on http://physics.stackexchange.com. I thank Christoph Ohm who almost managed to win my inertia of living always in the same apartment. I also thank Marco Roth for trying to convince us that one can simulate a Hydrogen molecule with two qubits, Daniel Zeuch for attending the group meetings even if he was sick all the times, and Jesse Slim for showing us beautiful videos of the Bloch sphere during those meetings. I thank Michael Voigt for the nice times we had in Maastricht and when we went out together. My thanks go also to Anirudh Krishna for the most expensive dinner I had in Aachen, and Xiaotong Ni for support in the update of my XBOX. I thank Anand Sharma for asking continuously when I was going to submit my thesis. If you are reading this it means I ﬁnally made it! I would also like to thank Daniel Weigend for very nice discussions about the GKP states and Kasper Duivenvoorden for nice discussions about "Better call Saul" instead. In addition, I would like to ofﬁcially apologize with Christophe Vuillot for the time he lent me his car and I had a car accident with it. I thank also Joel Pommerening, Florian Venn, Veit Langrock, Elias Walter and Yang Wang for being my patient students when I was a tutor for the course of Quan- tum Optics, and Mikhail Pletyukhov for teaching the course with me. I thank also Veit for stimulating discussions about alternative ways for reaching Mensa Vita. I wish also a good luck to the "newcomers" of the institute Evangelos Varvelis and Tobias Herrig. I thank, in particular, Evangelos for clarifying the subtle point that ix

Alexander the Great was greek and macedonian, but not from Macedonia. My special thanks go to the mother of all IQI members Helene Barton, who when I arrived patiently guided me through all the burocratic works, in which I am terrible at. I also thank Helene for being a constant point of reference for all the institute in these years and those to come. I thank all the people I met in Aachen in this period. I would like to begin by thanking my italian crew. In particular, I thank Martina Mariani for always listening to me in my difficult moments and Pamela Cacciatore for all the cakes she shared with me. I thank Gianluca Lipari for his dedication in doing homeworks at the german course, and Edoardo De "Nin" for our amazing NBA nights. I thank Michele Sambiagio and Magdalini Papadopoulou (zia Daniela) for developing the concept of "Schkatula". In addition, I thank Antonella Schiazza for showing to me that I can survive three hours of opera in german at the theater. I thank Amir Ahmadifar and Ravi Kanth for helping me cleaning the Symposion the day after my 28-th birthday. My thanks go also to my two tandem partners Rabea Werthmann and Stanislava Petkova, who tried somehow to teach me german. I thank Anna for spending with me the most beautiful times of my PhD period in Aachen, in Poland and in Italy. Vorrei ringraziare la mia famiglia a cominciare dai miei cugini (ma in pratica fra- telli) Alessandra e Francesco, la zia Daniela (non quella precedente), e lo zio Adria- no. Come promesso alla fine del mio PhD andremo finalmente a mangiare da Niko Romito! Ringrazio i miei nonni che tanto mi hanno insegnato nella mia vita. Ringra- zio anche la mia famiglia "belgese" di Genk, che mi ha fatto sentire a casa ogni volta che sono andato a trovarli. Ringrazio inoltre i miei amici storici "di giù" che ogni volta che torno, anche se non ci vediamo da mesi, a volte anni, mi mostrano sempre che nulla è cambiato nella nostra amicizia. Ringrazio papà che continua da lontano a darmi la forza di andare avanti sulla mia strada, anche nei momenti difficili. Infine, ringrazio la mamma per l’infinito supporto che mi ha dato e che continua a darmi nella mia vita, e l’amore spropo- sitato che riesce a trasmettermi ogni mattina, a 1468 chilometri di distanza, con un semplice "Buongiorno Alessandro!".

List of publications

This thesis is based on the following works

• B. Criger, A. Ciani and D. P. DiVincenzo, Multi-qubit joint measurements in circuit QED: stochastic master equation analysis. EPJ Quantum Technology 3, 6 (2016). Basis of chapter3.

• A. Ciani and D. P. DiVincenzo, Three-qubit direct dispersive parity measurement with tunable coupling qubits. Physical Review B 96, 214511 (2017). Basis of chap- ters3 and4.

• A. Ciani, B. M. Terhal and D. P. DiVincenzo, Hamiltonian quantum computing with superconducting qubits. Quantum Science and Technology 4 (3), 035002 (2019). Basis of chapters5 and6.

Further works not included in this thesis

• F. Hassler, J. Stubenrauch and A. Ciani, Equation of motion approach to black-box quantization: taming the multi-mode Jaynes-Cummings model. Physical Review B 99, 014515 (2019).

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Contents

Abstract iii

Zusammenfassungv

Acknowledgements vii

List of publications xi

1 Introduction1 1.0.1 Outline...... 6

2 Superconducting qubits and circuit QED9 2.1 Setting the stage...... 9 2.2 LC oscillator...... 10 2.2.1 Drives, coherent states and the importance of the non-linearity 13 2.3 The Josephson junction...... 16 2.4 Lagrangian of unconventional circuits...... 19 2.4.1 Transformer...... 20 2.4.2 Gyrator...... 20 2.5 General circuit quantization procedure...... 22 2.6 Main superconducting qubits...... 23 2.6.1 The Cooper-pair box and the transmon...... 23 2.6.2 Transmon as a Dufﬁng oscillator...... 28 2.6.3 Flux qubit...... 30 2.6.4 Fluxonium...... 32 2.7 Circuit QED: artiﬁcial atoms coupled to microwave cavities...... 32

3 Direct parity measurement 35 3.1 Stabilizer measurements in a nutshell...... 35 3.1.1 Ancilla-based parity measurement...... 38 3.1.2 Alternative proposals for parity measurements...... 39 3.2 Dispersive readout of superconducting qubits...... 41 3.2.1 Signal-to-Noise Ratio and measurement rates...... 45 3.3 Direct dispersive parity measurement...... 47 3.3.1 Concept...... 47 3.3.2 Conditions for three-qubit parity measurement...... 47 3.3.3 Measurement rates and information gain...... 51 xiv

3.3.4 Quantum trajectories and stochastic master equation...... 54 3.3.5 Quantum trajectories for three-qubit parity measurement.... 59

4 Tunable Coupling Qubit for parity measurements 63 4.1 Quantum switch term for a transmon...... 63 4.2 Tunable Coupling Qubit...... 67 4.2.1 Purcell protection...... 71 4.2.2 Tunable coupling...... 73 4.3 TCQ for direct three-qubit parity measurements...... 73

4.3.1 Achieving zero χ12 ...... 77

4.3.2 Case with χ12 6= 0...... 79 4.3.3 Black-box approach...... 80

5 Hamiltonian quantum computing 85 5.1 Different models of quantum compuation...... 85 5.2 Feynman quantum computer...... 87 5.2.1 Clock representation...... 89 5.2.2 Kitaev construction and the local Hamiltonian problem..... 90 5.3 Hamiltonian quantum computing on a lattice...... 92 5.3.1 Quantum gate with a hopping particle...... 93 5.3.2 Lattice of hopping particles...... 95 5.3.3 Valid Hamiltonian...... 96 5.3.4 Hopping Hamiltonian...... 97 5.3.5 Effective Hamiltonian...... 97 5.4 Multi-qubit gates...... 100 5.4.1 CNOT...... 101 5.4.2 Toffoli gate...... 104 5.5 Back to Feynman...... 107 5.5.1 Peres’ trick...... 112 5.5.2 Multi-qubit gates in the new model...... 114 5.6 Some freedom in the model...... 116 5.7 Analysis of unitary errors...... 116 5.7.1 Perturbation theory analysis...... 117 5.7.2 Wavefront analysis and effect of disorder...... 118 5.7.3 Errors in the CNOT logic...... 121

6 Towards an implementation with superconducting qubits 125 6.1 Dual rail encoding...... 125 6.1.1 Loss of particles...... 126 6.2 Implementation with superconducting qubits...... 128 6.2.1 Cross-Kerr interactions...... 128 6.2.2 Hopping interaction...... 134 6.2.3 Estimates of parameters and trade-offs...... 136 xv

6.3 Full implementation...... 139 6.4 Main challenges and practical considerations...... 141

7 Conclusions 143 7.1 Outlook...... 145

A Transmission lines 147 A.1 Inﬁnite transmission line...... 147 A.2 Finite transmission line...... 150

B Universality of Hadamard, controlled-Hadamard and CNOT 155

Bibliography 157

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List of Figures

2.1 LC oscillator...... 10 2.2 Passive sign convention for voltage and current across one element... 11 2.3 LC oscillator with current source...... 14 2.4 Josephson junction...... 16 2.5 Transformer with turns ratio T...... 20 2.6 Gyrator...... 21 2.7 Example of an electrical network...... 23 2.8 Cooper-pair Box circuit...... 24 2.9 Energy levels of the CPB Hamiltonian Eq. 2.68 as a function of the

offset charge ng. The zero of the energy is taken to be in each plot

the minimum of the energies of the lowest level. In addition, E01 is

considered in each plot at the degeneracy point ng = 0.5...... 26 2.10 Relative and absolute anharmonicities for the Cooper-pair box as a

function of EJ/EC. The parameter ng is set to ng = 0.5...... 27

2.11 Peak-to-peak charge dispersion as a function of EJ/EC for the first 3 energy level differences...... 27 2.12 Charge dispersion for the first two levels vs. anharmonicity...... 28 2.13 Circuit and potential of the rf-SQUID flux qubit...... 30 2.14 Circuit for the three-junction flux qubit and the capacitively shunted flux qubit...... 31 2.15 Fluxonium circuit...... 32 2.16 Transmon coupled to a LC oscillator...... 33

3.1 Ancilla-based stabilizer measurements...... 39

3.2 Phase of the reflection coefficient as a function of ∆d = ω − ωr for the qubit in the excited (red line) and in the ground (blue line) state. In the plot χ = κ/2...... 43 3.3 Phase space evolution of the cavity field depending on the state of the qubit. This plot was obtained using the drive amplitude shown in Fig. 3.4 and χ = κ/2...... 43 3.4 Drive amplitude used to obtain Fig. 3.3. Taking a total measurement √ time of τ = 28/κ we set the following parameters ess = 0.5 κ, σ =

4/28τ, ton = 1/28τ, toff = 16/28τ...... 44 xviii

3.5 Phase space evolution of the output ﬁeld. The pulse used is the same as the one in Fig. 3.4, and the parity condition is assumed to be

matched. In particular, the parameters are κ1 = κ2 = κ, χ1 = χ2 =

χ = κ/2, χ12 = 0...... 51 3.6 a) Average parity information gain for a measurement time τκ = 28 as

a function of χ1/κ and χ2/κ. The measured quadrature of the output ﬁeld is always chosen to be the one that maximizes the information content about the state of the system. b) Missing parity information

on a semilog scale for the symmetric case χ1 = χ2 (blue solid line)

and the asymmetric case in which χ2 = 0.3κ (orange dashed line). In both cases, we see a minimum of the missing information (maximum

information gain) at χ1 ≈ κ/2...... 53 3.7 Information gain about the Hamming weight and the parity. In both cases neither relaxation nor pure dephasing of the qubits is considered. The parameters are the same as in Fig. 3.6. Note that curves lie on top of each other...... 54 3.8 Histogram of the ﬁdelities when the state identiﬁed from the measurement signal is |ψ−i. The histogram corresponds to 5031 trajectories. The simulations were generated taking χ to be all equal. The dephas-

ing rate is taken to be equal for all qubits and set as γz = χ/300, while the relaxation rate was set to zero. The measurement time is taken to be τχ = 10 with a time-dependent drive as in Eq. 3.21. In addition,

κ1 = κ2 = κ = 2χ...... 61

4.1 First three energy levels and transitions diagram for a transmon. The orange and green arrows denote the direct transitions that can be caused by the interaction with the resonator 1 and 2 respectively, when the transmon is coupled linearly to them. As we see the transitions are the same...... 64 4.2 Example of transition structure that would give zero quantum switch term...... 67 4.3 Basic circuit of the TCQ. The Josephson junctions can be substituted by ﬂux tunable SQUID loops allowing for the control of the Josephson

energies EJ±...... 68 4.4 First 6 energy levels of the TCQ Hamiltonian Eq. 4.17 as a function

of the offset charges ng+ and ng− (m = {0, 1, 2, 3, 4, 5}). The zero of the energy is taken to be in each plot the minimum of the energies of the lowest level. The transmons are considered symmetric in this case

and EI /EC = 1. The plots are obtained by direct numerical diagonal-

ization of HTCQ by writing it in the charge basis and truncating the Hilbert space...... 69 xix

4.5 First six energy levels and transitions of a TCQ. The states in which the qubit is encoded are shown in red. Again we use the convention that the first excited state is the logical zero state, while the ground the logical one state...... 71 4.6 First six energy levels and transitions of a TCQ. The states in which the qubit is encoded are shown in red...... 72 4.7 A) coupling strength to the resonator and B) frequency of the qubit as a function of the Josephson energies in a TCQ. We assumed the bare capacitances of the two transmons to be equal...... 73 4.8 First six energy levels of a TCQ with transitions that can be caused by the two resonators. In the general case the resonators can cause the same transitions...... 75 4.9 Transitions that give zero quantum switch and fixed coupling between the two resonators...... 78 4.10 Possible three-qubit parity measurement setup achieving zero quantum switch term...... 79 4.11 TCQ with the two bare transmons coupled capacitively only to one resonator. The capacitances of the bare transmons are taken to be equal for simplicity...... 79 2 4.12 χ1χ2 and χ12 as a function of the coupling capacitance CI. The data are shown taking EC as a unit (¯h = 1)...... 80 4.13 Possible configuration that allows a three-qubit parity measurement 2 with χ1χ2 ≥ χ12, with χ12 6= 0 using the modes n = 2 of two transmission line resonators...... 81 4.14 Black-box parity network...... 82

5.1 Example of a single-qubit gate...... 93 5.2 Examples of connected and disconnected strings of particles. The red dots denote the position of the particles...... 95 5.3 Quantum walk of the connected string...... 99 5.4 The CNOT gate...... 101 5.5 Direct Toffoli gate...... 104 5.6 Example of depth 2 quantum circuit with 3 qubits and gates executed in a snake-like order...... 108 5.7 Examples of connected and disconnected strings of particles in the new lattice model. The red dots denote the position of the particles... 109 5.8 String motion...... 109

5.9 Success probability PL(t) for L = 20 and L = 100 as a function of Jt... 111 5.10 CNOT in the ‘single-clock snake’ lattice model of Fig. 5.7. The green edges are those who need to be modiﬁed similar to what we do in Subsec. 5.4.1...... 115 xx

5.11 Freedom in the realization of the model. In (A) different colors represent different attractive interactions, while in (B) different colors denote different on-site energy...... 116

5.12 Time averaged probability for the strings to be disconnected PD as a function of time for different lattice sizes. The inset shows the scaling

of the steady state probability PD,ss (evaluated at the ﬁnal time) with N.117 5.13 Average position of the particle on the central track as a function of time for different lattice sizes N. The dashed horizontal lines identify the maximum value that can be reached for each lattice size...... 118 5.14 Screenshots of wavefront at different times. The bars represent the probability of ﬁnding a particle in the corresponding position...... 119 5.15 Ideal vs. disordered time evolution. The disordered data are averaged over 50 simulation runs. The faded areas represent the standard deviation of the disordered curves...... 120 5.16 Ideal long-time time evolution vs. on-site disordered long time evolution...... 121 5.17 Example of probability of success (blue line, left axis) and error (red line, right axis) as a function of time for J/∆ = 1/10...... 121 5.18 Time-averaged probability of error as a function of J/∆ for a CNOT after Jt = 3. The error scales with (J/∆)4...... 122 5.19 Long time behaviour for the average probability of error and success.. 123

6.1 Cross-Kerr coupler between two transmons on different adjacent tracks. The different color of the transmons denotes that they can have different frequencies...... 129 6.2 Capacitive coupling between two transmons on the same track. The fact that the transmons have same color denotes that they have the same frequency...... 135

6.3 Total potential Utot(Φ1, Φ2) for the parameters in Table 6.1. α is set to 0.042...... 136 6.4 Hopping (a) and cross-Kerr (b) coupling parameters for α close to the condition γ = 0. We see that the hopping coupling is zero at α ≈ 0.043, corresponding to a Josephson energy of the small junction equal bare to αEJ = 51.72 in units of EC ...... 137 6.5 Two equal transmons (red) coupled to a common transmon (green) via the coupler analyzed in Subsec. 6.2.1. Parameters are not shown for simplicity...... 138 6.6 Layout concept for the Hamiltonian quantum computing scheme analyzed in this chapter with superconducting qubits...... 140 6.7 Examples of circuits for implementing single-qubit gates...... 140 6.8 Circuit for realizing the strong ZZ attractive interactions...... 140 6.9 Layout concept for the Feynman lattice...... 141 xxi

A.1 Discrete model of a inﬁnite transmission line...... 147

B.1 Implementation of a Toffoli gate using an ancilla qubit, a CNOT and controlled-controlled-iY and controlled-controlled-(iY)† gates...... 155 B.2 Implementation of a controlled-controlled-iY gate using CNOT, controlled- Hadamard and controlled-Z gates. The implementation of controlled- controlled-(iY)† follows analogously...... 155 B.3 Controlled-Z gate from Hadamard and CNOT...... 156

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List of Tables

4.1 Parameters for Fig. 4.12. We assumed equal transmons CJ+ = CJ− =

CJ, EJ+ = EJ− = EJ and also equal resonators C1 = C2 = Cr, L1 = 2 2 2 L2 = Lr. LJ = Φ0/(4π EJ), EC,bare = e /(2CJ). These are realistic parameters considering the resonators to be transmission line resonators

with characteristic impedance Z0 = 50 Ω. Additionally, all necessary approximations are well satisﬁed...... 80

bare 6.1 Parameters in units of EC ...... 135 6.2 Estimates of typical parameters achievable in our implementation... 139

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List of Abbreviations

AQC Adiabatic Quantum Computation BCH Baker-Campbell-Hausdorff CPB Cooper-Pair Box CSS Calderbank -Shor-Steane FQHE Fractional Quantum Hall Effect HQC Hamiltonian Quantum Computation IQI Institute for Quantum Information LNN Linear Nearest-Neighbour QA Quantum Annealing QEC Quantum Error Correction QMA Quantum Merlin Arthur RWA Rotating Wave Approximation SME Stochastic Master Equation SNR Signal-to-Noise Ratio SW Schrieffer-Wolff TCQ Tunable Coupling Qubits TLS Two-Level System

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A mamma e papà, come sempre. . .

Chapter 1

Introduction

The ﬁeld of quantum information and quantum computation has arisen during the eighties and its birth was mainly motivated by the following ideas. With the advent of classical computers and with the development of the theory of computational complexity, physicists started to realize that quantum systems are hard to simulate with classical resources. In quantum systems, a linear increase of the number of subsystems leads to an exponential increase of the size of the problem. To make a concrete example let us consider a system composed of N two-level systems (TLS), which we will also refer to as qubits. According to the rules of quantum mechanics the dimension of the total Hilbert space is then 2N. This means that in order to study this system we need to deal with matrices of dimension 2N × 2N. As a simple experiment we can try to see what happens when we try to store a matrix of this size on our laptop. The following simple Python script tries to store an identity matrix called ”pluto” of dimensions 215 × 215 and an identity matrix called ”pippo” of dimensions 216 × 216:

1 import numpy as np pluto= np.identity(2∗∗15) 3 del pluto pippo=np. identity (2∗∗16)

5 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

MemoryError Traceback(mostrecent call last ) 7 in () −−−−> 1 pippo= np.identity(2∗∗16) .

While there is no problem with the identity matrix pluto, the system is not able to store the matrix pippo. This simple experiment sets the limit of the number of two-level systems that can be simulated on my seven year old laptop with 6GB of RAM to 15. There might be some more clever way to limit the RAM needed, but at any rate we cannot get too far away from this number, if we are not willing to sacrifice computational time, at least with my laptop! It is estimated that the largest 2 Chapter 1. Introduction number of qubits that can be simulated on the most powerful classical supercom- puters available in the world at the time of this writing is ≈ 50 [1]. This number sets the limit for the so-called quantum supremacy. This means that if we are able to engineer a system composed of more than 50 qubits, in which we can also perform arbitrary operations, then there is no classical supercomputer in the world that is able to simulate that system. It is one of the current challenges to achieve the limit of quantum supremacy. It is actually debatable that achieving quantum supremacy in the previous sense has any deep fundamental meaning at a theoretical level, but it sets however a sort of break-even point between classical computers and what we will call quantum computers. We can say however that quantum supremacy has at least been achieved with respect to my laptop, since there exists quantum computers which implement 16 qubits [2]. The take-home message of this discussion is that quantum systems are hard to simulate with classical computers. It seems that to simulate quantum systems one needs other quantum systems to do it efficiently, and this was the main message of one of the Feynman’s papers on quantum computation [3]. There is another reason that has driven the field of quantum computation which is the idea that quantum bits (qubits) should be somehow more powerful than classical bits. A classical bit can assume two values that can be either 0 or 1, or if you want heads or tails, black or white, or whatever binary variables you can think of. We will denote these states by |0i, |1i, meaning that we associate to them two orthogonal vectors in a two-dimensional vector space " # " # 1 0 |0i = , |1i = . (1.1) 0 1

Denoting by |Ψi the state of a classical bit, we can have either |Ψi = |0i or |Ψi = |1i. Now, without any apparent reason, let us consider the abstract case in which the state |Ψi is not limited to have values |0i or |1i, but it can be in a linear combinations of them, meaning that we can write

|Ψi = c0 |0i + c1 |1i , (1.2)

2 2 with c0 and c1 complex numbers that satisfy a normalization condition |c0| + |c0| =

1. It is clear that we recover the classical bit if we restrict the values of c0, to be either

0 or 1, but given the infinite possibilities in the choice of c0 and c1, the classical case starts to look like a prison. For this reason, we are tempted to say that a computation based on states like the one in Eq. 1.2 might be more powerful than a classical one. A system whose state is represented by a vector like in Eq. 1.2 is what we call a qubit [4]. The reason why we consider these kinds of systems is simply that they exist in nature, and their existence is established by quantum mechanics. Driven by these ideas, and many others to be fair, scientists (not only physicists) started to develop the theory of quantum information and quantum computation Chapter 1. Introduction 3 which witnessed several breakthroughs during the nineties in a mix of surprisingly positive and negative results. A first negative result is that there is a little lie in what I said before. The accessi- ble information of a single qubit cannot be more than that of a single bit, and so we might say that a qubit is not more powerful than a bit, despite its apparently infinite possibilities in the choice of the coefficients c0 and c1. This result was actually known since the seventies [5]. The reason why this happens is that previously I omitted to say how the information in a qubit is extracted, which is a further defining feature of a qubit. In practice, information is extracted by measuring the qubit in the computational basis, i.e., |0i, |1i. If the qubit is in a generic state given by Eq. 1.2, the system 2 will be measured in the state |0i with probability |c0| , while in |1i with probability 2 |c1| . This is a quite limiting feature of a qubit, but again nature works in this way. A pessimistic reader would then assume that if a qubit is not more useful than a bit, it is of no use. An optimistic reader instead would notice that the fact that a qubit is not more useful than a bit does not imply that many qubits together cannot be more useful than many bits together. The optimistic reader would have a good point in saying that, otherwise we would not be here almost fourty years later talking about quantum computation. In fact, as I said, the nineties were quite prolific years for the theory of quantum computation and quantum information. The point of view of the optimistic reader was confirmed by the discovery of several quantum algorithms that were performing faster than any classical known algorithm. The landmark of these algorithms is undoubtly Shor’s factoring algorithm [6]. We have all learned how to write an integer number as a product of prime numbers in primary school. Fortunately enough, our teachers didn’t assign us a homework that required to factor too large numbers, otherwise we will be computing for the rest of our lives. In fact there is to date no classically known algorithm that allows to solve this problem efficiently, ı.e., in the language of computational complexity in polynomial time, including the one you were using in school. It is actually believed that there is none, although this is still just a conjecture. Shor’s algorithm shows that instead it is possible to factor an arbitrary large number efficiently with a quantum computer. The first consequence of this is that we could have spared some time doing our homeworks if we had a quantum computer. The second consequence is that being able to solve efficiently the factoring problem, implies the ability to break public-key cryptosystems efficiently, putting in jeopardy the security of these systems. However, quantum information is not only about hacking cryptosystems, although this feature raised a lot of attentions. There are also other quantum algorithms that give a speed-up compared to the best known classical algorithms, but not many give a speed-up as spectacular as Shor’s algorithm. It is in general difficult to come up with good algorithms even classically, and the challenge is even more difficult quantum mechanically, since the algorithm is also expected to perform better than any classical one, otherwise it would not be even considered. 4 Chapter 1. Introduction

However, despite the growing interest in the field due to the development of quantum algorithms and also the optimism of the founding fathers, it was soon realized that building a quantum computer based on states like the one in Eq. 1.2 is an unprecedented challenge both at a theoretical and engineering level, which in terms of technological difficulty might be considered comparable to nuclear fusion power generation. To let the reader grasp the difficulty let us make again an example. We said that a classical bit can be identified with whatever binary physical variable. Suppose you make the following game with a friend of yours, which without loss of generality will be called Bob. Bob asks you a question which has answer yes or no. For instance he could ask me "Do you think that we will build a quantum computer in the next 20 years?". You do not give the answer directly to Bob, but you say that you store the answer in a system that you can find in your apartment, for instance a coin. You then explain to Bob that if the coin is tails the answer is yes, while if heads no, and that he can come to check the answer whenever he wants. Unfortunately Bob is quite a busy guy, and can only come in a month to check the answer. You are quite sure however that this is no problem. You will keep the coin in a safe place, and unless unpredictable catastrophic events happen, you are quite sure that in a month the coin will still be there with the correct answer stored in it. We have just shown that we can easily build quite reliable classical bits out of almost any object in your apartment. Now let us try to repeat the previous experiment with a quantum bit. We actually tell Bob that the answer is not stored in the states |0i or |1i, but in √ one of the two fancy states |±i = (|0i ± |1i)/ 2. The answer is yes if he finds the system in the state |+i, while no if the system is in |−i. We assume that Bob knows a bit of quantum mechanics and is able to measure in this basis. In many cases, the states |0i and |1i are taken to be the ground state and the first excited state of some Hamiltonian respectively. In this case, the qubit is subjected to relaxation processes that bring whatever initial state |Ψi to the ground state. It is clear that if this hap- √ pens and we basically end up in the state |0i = (|+i + |−i)/ 2, Bob would get a completely randomized result from his measurement. For instance typical decay times of superconducting transmon qubits that we will treat in this thesis are usually 20 − 50 µs, and so it is better for Bob to hurry up if he wants to know the right answer. This silly example shows an important feature of quantum systems. General quantum states like the one in Eq. 1.2 are extremely fragile with respect to uncon- trolled noise, which causes a high error rate. We have just seen an example with the process of relaxation. However, this is only one kind of process. More generally a quantum system is subjected to decoherence, whose effect is to destroy its quantum power and to make it behave as a classical system [7]. The fragility of quantum states is what has set fundamental limits in the realization of quantum computers. Scientists tried to understand if it was possible to deal with noise and the associated errors, and somehow correct them. This gave rise to the theory of quantum error correction (QEC). The theory turns out to be quite Chapter 1. Introduction 5 beautiful and elegant, and it shows, in some aspects, deep connections with relatively recent topics in condensed matter physics, such as topological matter. Part of the theory, namely the stabilizer formalism, can also be interpreted as a generalization to the quantum world of the classical theory of linear codes. QEC is based on the same idea as classical error correction, that’s to say use many imperfect qubits to build a perfect one. It was also shown that not only we can protect a quantum state for memory purposes, like for instance storing the answer to Bob question reli- ably, but also to perform a quantum computation fault-tolerantly. This is possible at least if the error rate is below some threshold value that depends on the particular quantum error correcting code we are considering [4]. After the development of the theory many implementations of a quantum computer were proposed using quantum dots [8], trapped ions [9], nuclear magnetic resonance [10] and linear optics [11], just to mention some of them. Also some specific criteria were formulated for a quantum system to be considered a good candidate for building a quantum computer, which have been known since then as the DiVincenzo criteria [12]. To date we cannot say that there is a clear winning candidate system to build a quantum computer, like there is for classical computers (your solid-state-based laptop). All implementations face their own problems with their main sources of noise and difficulties in controlling the quantum state. It is usually the case that the ability to control the quantum state easily is traded with higher sensitivity to noise. This thesis deals with one of the possible implementations of a quantum computer, which is the one based on so-called superconducting qubits [13], interest in which has grown exponentially since the first experiments showing coherent controls on macroscopic quantum systems at the end of the nineties [14]. The key feature of these systems is indeed the word ”macroscopic”. One usually associates quantum mechanics with the extremely small world, but it turns out that it is possible to observe quantum phenomena also with observables that we would normally call macroscopic such as voltages and currents in a circuit. It turns out that if the basic constituents of a system behave quantum mechanically, some macroscopic degrees of freedom can also survive the statistical averaging and behave quantum mechanically. With this I do not mean just to show the quantization of some property, but really that in order to describe them we need to use states like the one in Eq. 1.2, usually with more than two levels. The reason why this happens in these systems is the phenomenon of superconductivity [15]. There are two main features of superconductors that are the key to success of superconducting qubits. First of all, the system is, as the word says, superconducting, which means that it has a very small dissipation (in principle zero at DC). This means automatically relatively low noise. Another feature is the existence of a special circuit element, namely the Josephson junction, which adds to simple capacitances and inductances as the basic circuit elements that we can use to build our circuits. The Josephson junction is paramount due to the fact that it is a non-linear element. If we didn’t have available a non-linear 6 Chapter 1. Introduction element we would be left only with a sad collection of harmonic oscillators which will always be stuck in quasi-classical states, and cannot show the true quantum power. A further appealling feature of superconducting quantum systems is the possibility to engineer the parameters of the problem by design. We will see that in our language a capacitance will play the role of a mass. By changing its value we can then obtain systems with different ”masses”. This means that we are not forced to work with a fixed, god given mass like the mass of the electron. Different parameter regimes may be achieved by design. In a certain sense this has also a downside since fabrication inaccuracies affect the properties of our artificial atoms, posing a problem of reproducibility, while we are quite confident that real atoms are all equal to each other. Last but not least, there is another important practical aspect of quantum computation with superconducting qubits that is worth mentioning. Superconduct- ing qubits are somehow the systems proposed for quantum computing that more closely resemble the way a classical computer is built and that shares more overlap with the fields of electrical engineering. The field benefits from many synergies from microwave engineering and quantum optics. The further application of Josephson junction circuits for quantum-limited measurement and amplification also raised a lot of interest in these kinds of circuits [16].

1.0.1 Outline

The thesis is organized as follows. In Chap.2 we present the general formalism of circuit quantization needed for obtaining the Lagrangian and Hamiltonian of superconducting circuits. In particular, we highlight, providing mathematical justifica- tion, the importance of the existance of a non-linear element such as the Josephson junction. After an introduction to the Josephson effect, we also describe the main kinds of superconducting qubits and briefly the field of circuit QED. In Chap.3, we analyze theoretically a scheme for performing direct three-qubit parity measurements with superconducting qubits, first described in Ref. [17]. We provide a brief introduction to these kinds of measurements, and in general to stabilizer measurements, in the context of quantum error correction. We then explain how single- and multi-qubit measurements can be performed using resonators via the so-called dispersive readout, and how we can even perform parity measurements with this readout technique. Using input-output theory, we obtain explicitly the condition that must be satisfied in order to realize a parity measurement, and show how the scheme can be analyzed using quantum trajectory theory. We em- phasize how the presence of qubit-state dependent coupling of the resonators has a harmful effect on the scheme. Building on these results, we discuss a possible realization in Chap. 4.1. In particular, we show how using tunable coupling qubits, which are qubits in the same family of the transmon, we can achieve the necessary flexibility for the implementation of a three-qubit parity measurement. We show that when operated in a certain Chapter 1. Introduction 7 configuration, the harmful coupling of the resonators can be completely cancelled. The system can also be operated in a simplified configuration in which the harmful terms are present, but tolerable. We then move in Chap.5 to the analysis of an alternative model of quantum computation, which performs the desired computation via a time-independent Hamil- tonian, as first envisioned by Feynman in Ref. [18]. The model we analyze was proposed in Ref. [19] and it is initially formulated in terms of a lattice of hopping particles, in which the information is encoded in their internal spin degree of freedom. After explaining the basics of the model, we extend the analysis of multi-qubit gates introducing a scheme for performing a direct Toffoli gate. We also discuss a new related model which maps unitarily to the Feynman Hamiltonian. We explain how for this model, and for small scale circuits, we can employ perfect state transfer techniques to make the computation more efficient. Focusing mainly on the original lattice model, we provide some numerical analysis of the effect of imperfections. In Chap.6 we propose a possible implementation of the models analyzed in Chap.5 based on superconducting transmon qubits. The problem can be in fact mapped to qubits using dual rail encoding. Within this mapping the goal is to engineer the time-independent Hamiltonian with the desired interactions, which are in the language of transmon qubits strong cross-Kerr (ZZ) interactions, and weaker flip-flop interactions. After explaining the additional sources of errors introduced by this encoding, we move to the analysis of how the above-mentioned interactions can be realized in a controlled way. To this end we discuss a new direct coupler for obtaining the strong cross-Kerr interactions between transmon qubits, which uses arrays of few Josephson junctions. In addition, we remark the fact that while the weaker flip-flop interactions can be more easily realized, at least a sign change is needed for achieving quantum universality. In our passive implementation, unitaries with complex matrix elements cannot be obtained, and consequently universality can be achieved with Hadamard and Toffoli gates, or with Hadamard, CNOT and controlled-Hadamard gates as described in AppendixB. We finally provides also estimates of typical parameters and identify the main challenges to realize this model. The conclusions are drawn in Chap.7 where we also provide an outlook.

Chapter 2

Superconducting qubits and circuit QED

2.1 Setting the stage

This chapter presents a short introduction to the field of quantum computation with superconducting qubits, and in general to the analysis of these kinds of devices which is not limited to applications to quantum information processing. The main goal of this chapter is to introduce the tool of circuit quantization and to describe the main kinds of superconducting qubits that have been developed during the years. As we explained in Chap.1 the field of quantum information with superconducting qubits is the art of engineering useful quantum effects basically using capacitances, inductances and an unconventional element, the Josephson junction. We should also add that microwave drives and flux biases should also be considered in the game, as well as distributed linear elements, ı.e., microwave transmission line resonators or microwave cavities. The reason why we used the word microwave is that the typical frequencies at which these systems are designed fall indeed in the microwave regime. Using these basic elements we can tailor the energy level structure and the interactions of our quantum systems. This is not the whole story. The level of noise should be sufficiently low so that not only the quantization of energy levels, but also coherent quantum effects, such as Rabi oscillations, should be observable [20]. This means that the system must be operated at extremely low temperatures of the order of millikelvin, which is also well below typical critical temperatures of superconductors. In practice, the superconducting chips are always placed in huge fridges which have several and different cooling stages to reach such low temperatures. The microwave regime allows to have energy separations that are larger than the characteristic thermal energy kBT at which the systems are operated, and in addition allows the use of standard classical electronics. In the next sections we will describe how to treat these elements within the La- grangian and Hamiltonian formalism necessary to describe quantum mechanical systems. We will mainly focus on relatively simple circuits and we will thus only sketch the general procedure for obtaining the Lagrangian of a generic circuit which 10 Chapter 2. Superconducting qubits and circuit QED Φ

L C

Φg = 0

FIGURE 2.1: LC oscillator. is well described in many excellent papers in the literature [21, 22, 23, 24, 25,1]. In the next section we start with the simplest system: the LC oscillator.

2.2 LC oscillator

In this section we start to introduce the Lagrangian and Hamiltonian formalism for the simple LC oscillator depicted in Fig. 2.1. The goal of the game is to construct a Lagrangian that reproduces correctly the equations of motion, ı.e., Kirchhoff’s law on currents and voltages. We will use the usual convention that the current passing through an element is taken to be positive in the direction in which the voltage decreases as shown in Fig. 2.2. Let us analyze the circuit in Fig. 2.1. We will immediately start to look in an abstract way at this circuit, which helps to understand more complicated circuits. In general, we can look at a circuit as a graph G = (V, E), in which we have vertices (or nodes) V connected by edges (or branches) E. At each edge we have an element with a certain current-voltage relation. In the circuit in Fig. 2.1, we have two nodes connected by two edges. At one edge we have an inductor L and at the other a capacitor C each with their fundamental current-voltage relations which read respectively dI V (t) = L l , (2.1a) l dt dV I (t) = C c . (2.1b) c dt The Kirchhoff’s law on voltages states that the algebraic sum of the voltages on the loops has to be zero. In our case, we have only one loop, and Kirchhoff’s law implies simply

Vl(t) = Vc(t) = V(t). (2.2)

On the other hand Kirchhoff’s law on currents requires that the algebraic sums of the currents at each node of the graph is zero. In the circuit in Fig. 2.1 the Kirchhoff’s law on the top node reads

Il(t) = −Ic(t), (2.3) 2.2. LC oscillator 11

FIGURE 2.2: Passive sign convention for voltage and current across one element. and actually the same equation is obtained for the node at the bottom, which shows that they are not independent. It is a general feature that a circuit with N nodes has only N − 1 independent Kirchhoff’s laws on currents. We now want to obtain Eqs. 2.2, 2.3 using a Lagrangian formalism. We start by defining two new variables for each branch Z t 0 0 Φb(t) = dt Vb(t ), (2.4a) −∞ Z t 0 0 Qb(t) = dt Ib(t ) (2.4b) −∞ where in our case b = l, c. Φb(t) and Qb(t) are interpreted respectively as branch fluxes and charges. Let us focus on the branch fluxes for the moment. It is clear that by definition the voltage on the branch is given by Vb(t) = Φ˙ b(t). Thus, Kirchhoff’s law on voltages Eq. 2.2 implies

Φc = Φl + const (2.5)

The constant in this specific case does not play any role, but in general it does when we consider circuits with non-linear elements like superconducting circuits with Josephson junctions. In general, Kirchhoff’s law on loops expressed in terms of branch fluxes states that the algebraic sum of the branch fluxes in a loop is equal to the flux enclosed in the loop. Eq. 2.5 is a holonomic constraint in the language of classical mechanics [26], which embeds automatically Kirchhoff’s law on voltages Eq. 2.2. We want to show now that we can find a Lagrangian L(Φ, Φ˙ ) whose associated Euler-Lagrange equation reproduces the Kirchhoff’s law on current Eq. 2.3. As we know from classical mechanics the Lagrangian is given by a kinetic part, involving the time derivatives of the dynamical variables, minus a potential which is a function of the variables themselves 1. Thus, in our case, a strong candidate Lagrangian is C 1 L(Φ, Φ˙ ) = Φ˙ 2 − Φ2, (2.6) 2 2L 1This is the case if we do not consider velocity-dependent forces such as the Lorentz force. 12 Chapter 2. Superconducting qubits and circuit QED where the first kinetic term is the energy stored in the capacitance, while the potential the energy stored in the inductance. In this picture, we see that the capacitance plays the role of a mass, while the inverse of the inductance that of the spring constant in a mechanical analogy. We will see that the roles can be inverted in this case, and in general for linear circuits. At any rate it is easy to show that the Euler- Lagrange equation that is derived from Eq. 2.6

d ∂L ∂L − = 0, (2.7) dt ∂Φ˙ ∂Φ is just the Kirchhoff’s law on current Eq. 2.3. From the Lagrangian we can obtain the Hamiltonian. The ﬁrst step is to identify the conjugate variable of the ﬂux Φ

∂L Q = = CΦ˙ , (2.8) ∂Φ˙ which in this case is equal to the (branch) charge on the capacitance. The Hamilto- nian is obtained as the Legendre transform of the Lagrangian and reads

Q2 Φ2 H(Φ, Q) = QΦ˙ − L(Φ, Φ˙ ) = + . (2.9) 2C 2L

Once we have the Hamiltonian, or better once we have identified the conjugate variables, we can quantize the theory via canonical quantization [27], which consists in promoting the variables to operators and imposing commutation relations between conjugate variables [Φˆ , Qˆ ] = ih¯ , (2.10) with the commutator between two generic operators Aˆ, Bˆ given by [Aˆ, Bˆ] = AˆBˆ − Bˆ Aˆ. For the sake of clarity we have denoted the operators with the hat symbol, but we will remove it from now on unless differently specified, since it will always be clear by context when a variable is a scalar or an operator. The fact that we have chosen the flux Φ as our position-like variable was a completely arbitrary choice. We could have as well chosen the branch charge, in which case we would have satisfied automatically Kirchhoff’s law on current, while Kirchhoff’s law on loops would be the Euler-Lagrange equation. The quantum Hamiltonian of the harmonic oscillator is readily diagonalized by the introduction of annihilation and creation operators r hZ¯ Φ = (a + a†), (2.11a) 2 r h¯ Q = i (a† − a), (2.11b) 2Z which satisfy bosonic commutation relations [a, a†] = 1 and where we introduced √ the characteristic impedance of the LC oscillator Z = L/C. The Hamiltonian in 2.2. LC oscillator 13 diagonal form becomes 1 H = h¯ ω a†a + , (2.12) LC 2 and has eigenstates |ni with equally spaced eigenenergies

1 E = h¯ ω n + , (2.13) n LC 2 √ with n ∈ N and ωLC = 1/ LC the characteristic frequency of the LC oscillator. The states |ni are usually called Fock states or number states, since they are eigenstates of the number operator a†a. We also say that if the system is in the state |ni it has n photons, which are the elementary excitations. The ground state |0i is also known as the vacuum, since there are no photons, and presents fluctuations in both variables, which originate essentially from the commutation relations. The LC oscillator is arguably the simplest circuit we can quantize. Nonetheless, we will see that the transmon qubit does not differ too much from a harmonic oscillator. As a final comment, let us look back at Fig. 2.1. In the previous discussion we have introduced the branch variable Φ. An alternative way of proceeding is to introduce node variables like Φg and Φ1 in Fig. 2.1, in terms of which our branch variable will be Φ = Φ1 − Φg. This shows that one of the two node variables can actually be set to an arbitrary value, which is a consequence of the fact that we have only one independent Kirchhoff’s law on currents. Thus, the node variable approach reduces exactly to the branch variable one in this case, if we set for instance Φg = 0, which we call in this case a ground node. This choice however does not mean that that node represents a physical reference ground. In the general case, in the node fluxes approach, a circuit with N nodes will give rise to a Lagrangian with N − 1 independent degrees of freedom, ı.e., the node fluxes [21].

2.2.1 Drives, coherent states and the importance of the non-linearity

The LC oscillator is the prototype of a linear circuit. Linear circuits, or in general linear systems, are by deﬁnition systems in which the equations of motion are linear. Considering lossless circuits for which a Lagrangian formulation is possible, a linear and passive system gives rise to quadratic Lagrangians and quadratic Hamiltonians. If we add active drives, such as a voltage or current source, this will be represented by a ﬁrst order term in the Lagrangian. To see this let us consider a time-dependent current source in parallel with our LC circuit like in Fig. 2.3. The effect of the current source is simply introduced in the Lagrangian of the LC circuit Eq. 2.6 by adding a term −I(t)Φ, which then leads to the Hamiltonian

Q2 Φ2 H(Φ, Q) = + + I(t)Φ. (2.14) 2C 2L 14 Chapter 2. Superconducting qubits and circuit QED

L C I

FIGURE 2.3: LC oscillator with current source.

In the mechanical analogy the current source is equivalent to a forcing term. The Hamiltonian in second quantized form, ı.e., in terms of annihilation and creation operators then reads 2 r hZ¯ H(t) = h¯ ω a†a + I(t) (a + a†). (2.15) LC 2

It is convenient to analyze the effect of the drive in the so-called interaction picture in which we take as the Hamiltonian H0 responsible for the evolution of the operators † H0 = h¯ ωLCa a. The Hamiltonian in the interaction picture is then [27] r iH t h −iH t h hZ¯ −iω t iω t H (t) = e 0 /¯ (H(t) − H )e 0 /¯ = I(t) ae LC + a†e LC . (2.16) I 0 2

Since this is a time-dependent Hamiltonian which does not commute with itself at different times the associated time evolution operator is a time-ordered exponential

Z t i 0 0 U(t) = T exp − dt HI (t ) . (2.17) h¯ 0

We can however try to understand the nature of this operator. To this end let us first introduce the concept of a coherent state for a harmonic oscillator. We first define the displacement operator [28]

† ∗ D(α) = eαa −α a, (2.18) with α ∈ C. It is easy to show that the displacement operator is unitary

D(α)D(α)† = 1. (2.19)

A coherent state |αi is obtained from the vacuum by applying a displacement operator D(α) |αi = D(α) |0i . (2.20)

From this deﬁnition we realize that the vacuum can also be viewed as a coherent state with α = 0. Coherent states are eigenkets of the annihilation operator with

2We omit the constant terms. 2.2. LC oscillator 15 eigenvalue α a |αi = α |αi . (2.21)

There are several properties of the coherent states that resemble the behaviour of a classical harmonic oscillator (see Ref. [28] for a broader discussion). For our purposes we just need to know that by applying another displacement operator to a coherent state we get another coherent state, which follows from the property

∗ D(α)D(β) = eiIm(αβ )D(α + β). (2.22)

Let us now consider the time-evolution operator in Eq. 2.17 at inﬁnitesimal times dt. We get

r i hZ¯ −iω t iω t U(dt) = exp − I(t) ae LC + a†e LC dt = D(µ(t)dt), (2.23) h¯ 2 where we deﬁned r i hZ¯ µ(t) = − I(t) eiωLCt. (2.24) h¯ 2 We conclude that if we start for instance in the vacuum, or in general in a coherent state, after an inﬁnitesimal time dt the Hamiltonian HI (t) generates another coherent state. The property Eq. 2.22 actually also tells us that we expect this to be true for every time. In particular, U(t) can be obtained as (see Chap. 4 of Ref. [29])

U(t) = eiλ(t)D(α(t)), (2.25) where the coherent state amplitude is given by

Z t α(t) = dt0µ(t0), (2.26) 0 and we also deﬁned the global phase factor

t t Z Z Z 0 λ(t) = −i ds ds0ε(s − s0)I(s)I(s0)eiωLC(s−s ), (2.27) 4¯h 0 0 with the asymmetric step function  1, t > 0,  ε(t) = 1/2, t = 0, (2.28)  −1, t < 0.

The main message that we have to draw from this discussion is that if we start with the Hamiltonian of a harmonic oscillator, which is a linear system, and apply a linear forcing term, and if the system is initially in a coherent state, then it will always remain in a coherent state. This is true also for a collection of harmonic oscillators in which case we would get tensor products of coherent states. Thus, with linear 16 Chapter 2. Superconducting qubits and circuit QED

S I S

(A) Graphical representation of a Josephson junction.

CJ EJ CJ EJ

(B) Josephson junction symbol.

FIGURE 2.4: Josephson junction. systems and linear drives we can never isolate a qubit from the states of the harmonic oscillator 3, nor can we get any entanglement. It is worth pointing out however that this does not rule out the harmonic oscillator as a system to implement a qubit [30, 31, 32], but a non-linearity in the model is needed.

2.3 The Josephson junction

We now introduce the Josephson junction which, as we said, is the element that provides the necessary non-linearity in superconducting circuits. We do not have to delve too much into the theory of superconductivity to understand the Josephson effect, at least for our purpose of analyzing superconducting circuits. We direct the reader to dedicated literature for a more thorough analysis of the phenomenon of superconductivity [15, 33, 34, 35]. The following brief discussion borrows from Ref. [23]. All we need to understand is the current-ﬂux relation for a Josephson junction. We provide however a simple derivation which helps in the understanding of the

3Not necessarily the Fock states. 2.3. The Josephson junction 17 basic phenomenon and also in the discussion of superconducting qubits. In a superconductor at low temperatures the ground state is composed of electrons which are paired so that they form a new, bosonic particle known as a Cooper-pair, which in standard superconductors is made up of two electrons with opposite spins. Being bosonic particles, Cooper-pairs can occupy the same quantum state and the ground state can be described by a single, global wave function. Let us now consider Fig. 2.4a. It represents two superconductors, say A and B separated by an insulating barrier. We denote by Nˆ A and Nˆ B the number of Cooper-pairs in superconductor A and B respectively. We have put the hat symbol to remark the fact that they have to be considered to be quantum mechanical operators. However, the number of

Cooper-pairs of the total system is ﬁxed NTOT = Nˆ A + Nˆ B and it is consequently a scalar. The insulating material that separates the two superconductors can be seen essentially as a potential barrier, which quantum-mechanically allows for tunneling of Cooper-pairs from one side to the other. Consequently, the difference of Cooper- pairs is still a quantum mechanical operator, and therefore we deﬁne the operator

Nˆ = (Nˆ A − Nˆ B)/2. Its spectrum is discrete and in the assumption that we can treat the system as having an inﬁnite number of electrons it is identiﬁed by the following eigenvalue relation Nˆ |ni = n |ni , (2.29) with n ∈ {0, ±1, ±2, . . . }. This means that the spectral representation of the operator Nˆ is +∞ Nˆ = ∑ m |mi hm| , (2.30) m=−∞ which induces the completeness relation

+∞ ∑ |mi hm| = I. (2.31) m=−∞

Assuming also that only one Cooper-pair can tunnel at a time, we model this phenomenon with the Josephson tunneling Hamiltonian

E +∞ ˆ J HJ = − ∑ |mi hm + 1| + h.c., (2.32) 2 m=−∞ with EJ the Josephson energy which depends on the details of the junction. Notice that we have neglected the fact that in principle there is always a capacitance associated with the Josephson junction, since for now we want to focus solely on the contribution due to the Josephson effect. Eq. 2.32 is de facto the same as the Hamil- tonian of a one dimensional tight-binding model, or also called a continuous-time quantum walk, which will appear again in Chap.5 in a completely different context. The eigenstates of the Hamiltonian Eq. 2.32 are easily obtained as plane waves

1 +∞ |ϕi = √ ∑ eimϕ |mi , (2.33) 2π m=−∞ 18 Chapter 2. Superconducting qubits and circuit QED where we can take ϕ ∈ [0, 2π) and which satisfy

Hˆ J |ϕi = −EJ cos ϕ |ϕi . (2.34)

Notice that with this choice of normalization the states |ϕi satisfy the following or- thogonality relation hϕ0|ϕi = δ(ϕ0 − ϕ), (2.35) which implies the completeness relation

Z 2π dϕ |ϕi hϕ| = 1 (2.36) 0

We now introduce the concept of phase operator. Let us consider the operator

+∞ ˆ UJ = ∑ |m + 1i hm| . (2.37) m=−∞

† It is easy to show that it is a unitary operator, ı.e., UJUJ = 1. As such we can always write it as iϕˆ Uˆ J = e , (2.38) where we introduced the hermitian operator ϕˆ. We can then write

Hˆ J = −EJ cos ϕˆ. (2.39)

We now deﬁne the ﬂux operator of a Josephson junction to be

Φ Φˆ = 0 ϕˆ, (2.40) 2π with Φ0 = h/2e the superconducting flux quantum. The operator Φˆ can be in- trepreted as the branch flux across the junction, as we will show. First of all let us show how to express the current through the junction. The current operator in the Heisenberg picture is given by dNˆ Iˆ = 2e , (2.41) dt where, since we assume the elementary charge to be positive, we have taken the convention that the current is positive when flowing from superconductor B to A. Using the Heisenberg equation of motion for a generic operator Aˆ

dAˆ i = [Hˆ , Aˆ], (2.42) dt h¯ J we obtain

Iˆ = Ic sin ϕˆ, (2.43) with the critical current Ic = 2πEJ/Φ0. Eq. 2.43 is known as the first Josephson relation. In order to show that the flux operator defined in Eq. 2.40 can be interpreted 2.4. Lagrangian of unconventional circuits 19 as a branch flux we want to show that if we apply a voltage V(t) to the junction, then

dΦˆ = V(t). (2.44) dt

To this end it is useful to obtain the following important commutation relation

[eiϕˆ , Nˆ ] = −eiϕˆ , (2.45) from which we deduce the commutation relation between the phase operator ϕˆ and the number of tunnelled Cooper-pairs operator Nˆ

[ϕˆ, Nˆ ] = i. (2.46)

Adding a voltage V(t) means to add a term in the Hamiltonian Hˆ V = 2eVNˆ and thus we consider the Hamiltonian

Hˆ = Hˆ J + Hˆ V = −EJ cos ϕˆ + 2eVNˆ . (2.47)

Using the commutation relation Eq. 2.46 we obtain the Heisenberg equation of motion for the phase operator in presence of a voltage

dϕˆ 2e = V, (2.48) dt h¯ which is the second Josephson relation. Notice that the deﬁnition Eq. 2.40 and Eq. 2.48 imply Eq. 2.44 as expected.

2.4 Lagrangian of unconventional circuits

Superconducting circuits are usually composed of combinations of inductances, capacitances and Josephson junctions. Mutual inductances are also included and their treatment is essentially the same as that of an inductance. There are however two additional linear lossless elements that are worth exploring, which are the transformer and the ideal gyrator. While there is practically no physical transformer in superconducting circuits, the transformer enters in some equivalent circuits such as the Foster representation of a linear, lossless, reciprocal multi-port network [36, 37], which is used in the context of black-box quantization [38, 39, 40]. We will see an application of Foster’s circuit in Subsec. 4.3.3. On the other hand, the gyrator is the most basic non-reciprocal element. It is a two-port element characterized by its impedance matrix. The gyrator has been recently studied within the Lagrangian formalism for superconducting circuits [41] and also an approach for obtaining the Lagrangian of complex circuits involving gyrators has been proposed in Ref. [42]. Non-reciprocal elements are very important in superconducting circuits, and current 20 Chapter 2. Superconducting qubits and circuit QED

I1 I2 Φ1 Φ2 T

V1 V2

Φg1 = 0 Φg2 = 0

FIGURE 2.5: Transformer with turns ratio T. designs of quantum computers require a massive use of them. However, the physical realization of non-reciprocal elements is usually quite bulky, and consequently limits the scalability of these systems. Ideas to reduce the size of these systems based on the quantum Hall effect have been analyzed in Refs. [43, 44, 45, 46, 47]

2.4.1 Transformer

The ideal transformer is depicted in Fig. 2.5. It is a two-port lossless element which relates the voltage and currents on its two ports. Its action is essentially to put a constraint between the voltages

V2 = TV1, (2.49) where the real parameter T is usually called turns ratio. Since the ideal transformer is a lossless element the power entering from port 1 has to be equal to the power going out from port 2 at any time which implies

V1 I1 = −V2 I2, (2.50) where the minus sign originates from our convention on the currents. Thus, an ideal transformer also constraints the currents

1 I = − I . (2.51) 2 T 1

Assuming that we write the Lagrangian in terms of node ﬂuxes, identiﬁed in Fig. 2.5 the constraint Eq. 2.49 translates into

Φ2 = TΦ1, (2.52) which has the effect of eliminating a node variable. The fact that Eq. 2.49 also implies Eq. 2.51 follows when we consider putting another system in one of the ports, for instance just a current source.

2.4.2 Gyrator

The ideal gyrator is a two-port, non-reciprocal, lossless element whose symbol is depicted in Fig. 2.6a. The fact that the gyrator is non-reciprocal implies that it has 2.4. Lagrangian of unconventional circuits 21

I1 I2

G V1 V2

(A) Ideal gyrator.

Φ1 Φ2

G L1 L2

Φg1 = 0 Φg2 = 0

(B) Two generic elements coupled via a gyrator.

FIGURE 2.6: Gyrator. non-symmetric admittance and impedance matrices [48]. The ideal gyrator can be can be characterized by its admittance matrix Y " # " #" # I 0 −G V ~I = 1 = 1 = YV~ , (2.53) I2 G 0 V2 where G is the gyration constant. Referring to Fig. 2.6a, the Lagrangian of a gyrator is G L = (Φ Φ˙ − Φ˙ Φ ). (2.54) gyr 2 1 2 1 2 We will now show that the Lagrangian Eq. 2.54 correctly reproduces the admittance matrix of the ideal gyrator. To this end we consider the circuit in Fig. 2.6b in which the two ports of the gyrators are attached to two completely generic networks, described by a Lagrangian L1 and L2. The total Lagrangian is

L = L1 + L2 + Lgyr. (2.55)

In particular, L1 is a function of Φ1, Φ˙ 1 and some other internal degrees of freedom

L1 = L1(Φ1, Φ˙ 1; ΦI1, Φ˙ I1), (2.56) and the same is valid for L2. The key point is to understand that the total current passing through the element with Lagrangian L1, as depicted in Fig. 2.6b, is given by d ∂L1 ∂L1 IL1 = − . (2.57) dt ∂Φ˙ 1 ∂Φ1 We just have to pay attention to the sign convention now. In fact, according to the convention of Fig. 2.6a, the current passing through the gyrator on the left side 22 Chapter 2. Superconducting qubits and circuit QED would be d ∂L1 ∂L1 I1 = −IL1 = − − . (2.58) dt ∂Φ˙ 1 ∂Φ1

Analogous discussion holds for I2 d ∂L2 ∂L2 I2 = −IL2 = − − . (2.59) dt ∂Φ˙ 2 ∂Φ2

Let us now consider the Euler-Lagrange equation obtained from the total Lagrangian Eq. 2.55

L = L1 + L2 + Lgyr. (2.60)

The Euler-Lagrange equation associated with Φ1 reads d Lgyr ∂Lgyr − I1 + − = 0, (2.61) dt ∂Φ˙ 1 ∂Φ1 which gives

I1 = −GV2, (2.62) since Φ˙ 2 = V2. Analogously the Euler-Lagrange equation associated with Φ2 gives

I2 = GV1. (2.63)

Compactly we obtain Eq. 2.53 as expected. There is a clear mechanical analog of the Lagrangian of a gyrator Eq. 2.54. It has in fact exactly the same form as the Lagrangian term of a charged particle in a constant magnetic ﬁeld, where the gyraton constant G plays the role of the magnetic ﬁeld.

2.5 General circuit quantization procedure

In this section we briefly sketch the general procedure for obtaining the Lagrangian of an arbitrary superconducting network based on the node fluxes method (see also Ref. [25] for an alternative approach). We refer to related literature for a more rigorous and systematic treatment [21, 24, 22]. We consider circuits involving capacitances, inductances and Josephson junctions. We also allow the presence of static fluxes in the loops of the network. An electrical circuit can be described as a graph with nodes and edges (branches). We see an example in Fig. 2.7. In each branch we have a circuit element, in our case a capacitance, an inductance or a junction. In the node variable approach for obtaining the Lagrangian of the network the general recipe is the following:

1. to each node of the network associate a node ﬂux variable. In Fig. 2.7 we have a network with 5 nodes; 2.6. Main superconducting qubits 23

Φ2 Φ3

Φe2

Φ1 Φ4

Φe1

Φg = 0

FIGURE 2.7: Example of an electrical network.

2. choose arbitrarily one of the nodes of the network as reference node, or ground node, and set its node ﬂux variable to zero. In Fig. 2.7 we chose the grey node to be the ground;

3. identify a spanning tree, that’s to say a set of paths that from the ground node leads to all other nodes. The spanning tree is not unique and the red one in Fig. 2.7 is only an example (see also Fig. 2.1). To avoid problems, the spanning tree should be chosen such that it contains only capacitances if possible. The branches that are not included in the spanning tree are called closure branches (blue branches in Fig. 2.7). To each closure branch there is associated an irreducible loop that closes the spanning tree;

4. identify the branch variables on the spanning tree as Φb = Φi − Φj, where Φi

and Φj denote the node ﬂuxes connected to the branch. Write the corresponding term in the Lagrangian in terms of node ﬂuxes;

5. from ﬂuxoid quantization [15] the branch variables on the closure branches

must satisfy Φb = Φi − Φj + Φext,b + 2πn, with Φext,b the ﬂux enclosed in the irreducible loop associated with the closure branch and n ∈ N. Write the corresponding term in the Lagrangian in terms of node ﬂuxes.

2.6 Main superconducting qubits

2.6.1 The Cooper-pair box and the transmon

The Cooper-pair box (CPB) is historically one of the ﬁrst examples of a superconducting circuit that exhibits quantum mechanical behaviour and can be used as a qubit [49, 14]. Since most of this thesis is based on qubits closely related to the CPB we discuss it in some detail here. Its basic circuit is depicted in Fig. 2.8 in which we 24 Chapter 2. Superconducting qubits and circuit QED

Cg Φ

+ Vg − CJ EJ

FIGURE 2.8: Cooper-pair Box circuit. notice that it is essentially nothing more than a Josephson junction in parallel with a capacitance coupled capacitively to a voltage source. This circuit can be readily quantized. First of all, we take the node ﬂux Φ as in Fig. 2.8 and write the La- grangian of the circuit in terms of Φ and its time derivative Φ˙ . This gives

2 2 Φ˙ (Φ˙ − Vg) 2π L(Φ, Φ˙ ) = CJ + Cg + EJ cos Φ . (2.64) 2 2 Φ0

Introducing the conjugate variable of the ﬂux as

∂L Q = = (CJ + Cg)Φ˙ − CgVg, (2.65) ∂Φ˙ the Hamiltonian of the system is simply obtained as the Legendre transform of the Lagrangian:

2 (Q − CgVg) 2π HCPB(Φ, Q) = QΦ˙ − L(Φ, Φ˙ ) = − EJ cos Φ , (2.66) 2CΣ Φ0 with CΣ = CJ + Cg. The conjugate variable Q has clearly the interpretation of the charge on the island. In order to quantize the circuit we promote variables to operators and set the commutation relation between conjugate variables as usual

[Φ, Q] = ih¯ , (2.67) where Φ and Q should be taken as operators from now on. We also rewrite the Hamiltonian in terms of the the number of tunnelled Cooper pairs N = Q/2e and the superconducting phase ϕ = 2πΦ/Φ0 as

2 HCPB = 4EC(N − ng) − EJ cos ϕ, (2.68) where we introduced the charging energy

e2 EC = , (2.69) 2CΣ 2.6. Main superconducting qubits 25 and the reduced gate charge CgVg n = . (2.70) g 2e It is worth pointing out that ϕ has to be taken to be a compact variable, while its generator N has a discrete spectrum. ϕ and N are related to each other as an angle is related to the angular momentum. The behaviour of the Cooper-pair box is governed by two competing energy scales: the charging energy EC and the Josephson energy EJ. In addition, we see that the Hamiltonian Eq. 2.68 depends on the classical parameter ng and so will the energy levels. It can be shown analytically that the energy levels are periodic with period 1 with respect to the parameter ng. This is a consequence of Eq. 2.45, from which we deduce that e−iϕ Ne+iϕ = N + 1. (2.71)

Since eiϕ is a unitary operator, which also does not modify the periodic boundary condition, the spectrum does not change if we replace N → N + k with k ∈ Z. However, we expect that noise in the voltage source would cause randomized variation of the energy levels, which leads eventually to dephasing [23]. This kind of noise is called charge noise and it was a major problem of CPB and a limiting factor in achieving high coherence times in the ﬁrst realizations. We can already try to draw some qualitative conclusions about the expected behaviour of the CPB . In particular:

• in the limit EJ EC we expect the eigenstates of the Hamiltonian of the CPB to be approximately the charge states;

• in the limit EJ EC we expect that the dependency of the energy levels on ng decreases, hence the CPB becomes immune to charge noise;

• however in the limit EJ EC we also expect the energy levels of the CPB to resemble more and more those of a harmonic oscillator with equal spacing between the levels, which of course hinders the possibility to isolate a two-level system. The reason for this is that the amplitude of the cosine increases and thus, locally, it looks more and more like the potential of a harmonic oscillator.

The last two considerations are crucial for the discussion of the transmon, that basically works in a regime of parameters, in which there is a compromise between sensitivity to charge noise and anharmonicity.

We plot the energy levels of the CPB as a function of ng in Fig. 2.9. The results are obtained numerically by diagonalizing in the charge basis. However, the Hamil- tonian Eq. 2.68 has an analytical solution in terms of Mathieu functions [50]. In Fig. 2.9, we can observe the periodicity of the energy levels with period 1 as a function of the redued gate charge ng. Furthermore, we notice that, as expected from our qualitative predictions, increasing the ratio EJ/EC decreases the dependency of the energy levels on the offset charge, although the system becomes more and more 26 Chapter 2. Superconducting qubits and circuit QED

EJ/EC=1.0 / =5.0 10 EJ EC 3.0

2.5 8

2.0 6

01 1.5 01 / E / E m 4 m 1.0 E E

0.5 2

0.0

0 -0.5 -2 -1 0 1 2 -2 -1 0 1 2

(A) (B)

EJ/EC=10.0 EJ/EC=50.0 3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5 01 01 / E / E

m 1.0 m 1.0 E E

0.5 0.5

0.0 0.0

-0.5 -0.5 -2 -1 0 1 2 -2 -1 0 1 2

ng ng

FIGURE 2.9: Energy levels of the CPB Hamiltonian Eq. 2.68 as a function of the offset charge ng. The zero of the energy is taken to be in each plot the minimum of the energies of the lowest level. In addition, E01 is considered in each plot at the degeneracy point ng = 0.5. harmonic. In order to evaluate quantitatively this behaviour we can mathematically deﬁne what we mean by anharmonicity and dependency of the levels on the reduced gate charge ng. The absolute anharmonicity δ and relative anharmonicity δr are deﬁned as

δ = E12 − E01, δr = δ/E01, (2.72) where E12 = E2 − E1 and E01 = E1 − E0. In general, we denote Eij = Ei − Ej. In order to quantify the dependency of the energy levels on ng, we deﬁne the peak-to-peak charge dispersion ei,j for the levels i and j as

ei,j = Eij(ng = 1/2) − Eij(ng = 0). (2.73)

As shown in Fig. 2.10a, and as expected, the relative anharmonicity tends to go to zero as the ratio EJ/EC is increased and has negative values for EJ/EC higher than approximately 10, with a minimum between 15 and 20. Fig. 2.10b tells us also that for high EJ/EC, δ ≈ −EC, a result that we will soon analytically justify. The negative value of the anharmonicity means that the transition 1 ↔ 2 has smaller frequency than the transition 0 ↔ 1.

Fig. 2.11 shows that the energy levels tend not to depend on the parameter ng as EJ/EC becomes large, a result that we have already predicted and qualitatively observed with the ﬂattening of the energy levels in Fig. 2.9. Finally in Fig. 2.12 we can compare the two effects of loss of anharmonicity and immunity to charge noise 2.6. Main superconducting qubits 27

0.8 3

0.6 2

0.4 1 C r δ δ / E

0.2 0

0.0 -1

-0.2 -2 0 20 40 60 80 0 20 40 60 80

EJ/EC EJ/EC (A) (B)

FIGURE 2.10: Relative and absolute anharmonicities for the Cooper- pair box as a function of EJ/EC. The parameter ng is set to ng = 0.5.

1 = 1 / 2 )

g 0.001 ϵ01 ( n

01 ϵ12 |/ E

ij ϵ23 | ϵ

10-6

0 20 40 60 80

EJ/EC

FIGURE 2.11: Peak-to-peak charge dispersion as a function of EJ/EC for the ﬁrst 3 energy level differences. 28 Chapter 2. Superconducting qubits and circuit QED

0.01

|ϵ01|/E01(ng=1/2)

δr(ng=1/2)

10-5

10-8

0 20 40 60 80

EJ/EC

FIGURE 2.12: Charge dispersion for the ﬁrst two levels vs. anharmonicity.

as EJ/EC increases. While the charge noise is exponentially suppressed, the anharmonicity decreases only slowly. This means that we can work in a regime where the CPB is insensitive to charge noise, while retaining a sufﬁcient anharmonicity to be able to drive the desired qubit transition, without exciting the other levels. This is the regime in which the transmon works (and indeed deﬁnes what a transmon is), which we can roughly identify with the regime 10 ≤ EJ/EC ≤ 100 [51]. Finally, it is worth pointing out that the fact that we are able to isolate a qubit transition does not mean that we can completely forget about the other levels treating the system as a two-level system. In fact, at least also the second excited level should be considered in the analysis, since it gives a non-negligible effect when considering the coupling with other systems such as microwave resonators.

2.6.2 Transmon as a Dufﬁng oscillator

In this subsection we show how we can approximately treat the transmon regime of a CPB as a Dufﬁng oscillator [52]. This approach will be very important also for the remaining part of the thesis. First of all, since in the transmon regime the energy levels are independent of the reduced gate charge ng we can omit this parameter. Thus, we rewrite the Hamiltonin 2.68 as 2 H = 4EC N − EJ cos ϕ. (2.74)

The idea at this point is just to expand the cosine potential up to 4-th order and then proceed with the standard second quantization of a harmonic oscillator. This treatment builds on the physical intuition given in Subsec. 2.6.1, since we expect the system to behave more and more as a harmonic oscillator as EJ/EC is increased, but 2.6. Main superconducting qubits 29 with some ﬁnite anharmonicity. The 4-th order term in the expansion of the cosine is the term of higher order that breaks the harmonicity of the system, and thus it makes sense to think that such an expansion may lead to a good model for the transmon. Carrying out this Taylor expansion, and of course neglecting constant terms, we get

E E H = 4E N2 + J ϕ2 − J ϕ4 + O(ϕ6). (2.75) T C 2 24

We will omit the O(ϕ6) from now on. At this point we diagonalize the harmonic oscillator part as usual, introducing annihilation b and creation b† operators. In particular, M = 1/(8E ) plays the role of the mass in the system, while k = E plays the C J √ role of the spring constant, and consequently the natural frequency ω0 = k/M = p 1/¯h 8EJ EC. Hence, in complete analogy with Eqs. 2.11 we introduce annihilation and creation operators for the transmon as

2E 1/4 ϕ = C (b + b†), (2.76a) EJ

i E 1/4 N = J (b† − b). (2.76b) 2 2EC Substituting Eqs. 2.76 into Eq. 2.75, we get p 1 EC H = 8E E b†b + − (b† + b)4. (2.77) T J C 2 12

Keeping only diagonal terms in the expansion of (b† + b)4, ı.e., non-rotating terms, and applying iteratively the commutation relation, we obtain

(b† + b)4 ≈ 6b†b†bb + 12b†b. (2.78)

Hence, we ﬁnally write

δ H ≈ H = h¯ ω b†b + h¯ b†b†bb, (2.79) T Duff t 2 p where ωt = 1/¯h( 8EJ EC − EC) and δ = −EC/¯h. The Hamiltonian Eq. 2.79 is that of a so-called Dufﬁng oscillator. We have also obtained that the energy difference between ﬁrst excited state and ground state, which are the states that will be used as qubit, is approximately given by ωt, while the anharmonicity, as noticed in Subsec. 2.6.1 is basically equal in modulus to the charging energy. It is worth pointing out that the Hamiltonian Eq. 2.79 is in diagonal form and thus, we can immediately obtain all the energy levels. 30 Chapter 2. Superconducting qubits and circuit QED

ϕext = π

CJ, EJ Φext L

|Li| Ri

(A) (B)

FIGURE 2.13: Circuit and potential of the rf-SQUID ﬂux qubit.

2.6.3 Flux qubit

The flux qubit is like the CPB one of the first proposed superconducting qubits [53, 54]. Flux qubits have witnessed considerable evolution compared to their initial proposals and we can say that there are many circuits that deserve to be included in the family of flux qubits. Here, for simplicity we focus on the simplest circuit that conveys the main idea which is that of a rf-SQUID flux qubit [55]. This is also the kind of qubit that is used in the D-Wave architecture [56, 57]. The circuit is shown in Fig. 2.13a and it is essentially a Josephson junction shunted by an inductance, where we also see the presence of an external flux in the inductive loop. We readily obtain the Hamiltonian of the rf-SQUID flux qubit in terms of the superconducting phase difference across the inductance and its conjugate variable

2 H = 4EC N − U(ϕ), (2.80) with potential E U(ϕ) = E cos(ϕ + ϕ ) − L ϕ2, (2.81) J ext 2 2 2 with ϕext = 2πΦext/Φ0 and we have defined the inductive energy EL = Φ0/(4π L). In Eq. 2.80 we notice that compared to the CPB we do not have a reduced gate charge ng. The reason for this is that the variable ϕ should not be taken as a compact variable anymore, essentially because the inductive shunt connects the two islands. Consequently, the variable N does not have a discrete spectrum and in this case ϕ and N should be interpreted as position and momentum, rather than an angle and an angular momentum. In this case, the reduced gate charge does not influence the energy levels and can be removed completely by a unitary transformation 4. This implies that the rf-SQUID flux qubit is insensitive to charge noise. How the system interpolates between a compact and a non-compact behaviour in the limit of L → +∞ is not trivial and has been studied in Ref. [58].

4 The same unitary transformation would remove the ng from the Hamiltonian in the compact case, but ng would appear again in the boundary conditions. 2.6. Main superconducting qubits 31

Φext

FIGURE 2.14: Circuit for the three-junction ﬂux qubit and the capacitively shunted ﬂux qubit.

The rf-SQUID ﬂux qubit is usually operated close to ϕext = π and with β =

EJ/EL > 1, in which case the system shows a double-well potential, which is qualitatively depicted in Fig. 2.13. We can identify localized states in each well by expanding the potential around each of the two minima, and taking ideally the limit of infinite barrier. In particular, we denote the left and right ground states by |Li and |Ri, and we choose them as our computational basis states. They correspond to left and right circulating supercurrents in the loop. They are not however eigenstates of the Hamiltonian since the presence of a finite barrier allows tunneling between the two wells. The first two eigenstates are approximately symmetric and antisym- √ metric superpositions of the states |Li and |Ri, ı.e., (|Li ± |Ri)/ 2. Restricting to the first two levels and writing the Hamiltonian in the computational basis of the rf-SQUID flux qubit, we get σz H = ε + ∆σx, (2.82) FQ 2 where both ε and ∆ depend on the parameters of the problem EJ, EC and EL, as well as on the external flux. In our example, at ϕext = 0 we have ε = 0. The energy levels are quite sensitive to the external flux and as a consequence flux qubits are sensitive to flux noise. Notice that, compared to the transmon, flux qubits are much more anharmonic. The rf-SQUID that we have briefly described is only an example of the family of flux qubits. The most common implementation is the three-junction flux qubit depicted in Fig. 2.14, proposed in Ref. [54]. In this circuit, the inductance of the rf-SQUID flux qubit is replaced by two identical Josephson junctions in series. Im- portantly the two added Josephson junctions are bigger than the other junction, ı.e., they have larger characteristic Josephson and charging energies 5. The reason for using junctions in place of inductances is to reduce the size of the inductive loop and moderate the problem of flux noise. Another evolution of the flux qubit is the capacitively shunted flux qubit [59], which is obtained from the three-junction flux qubit just by adding a large capacitance in parallel with the small junction. These improvements, as well as improvements in fabrication and a better understanding of noise sources, improved dramatically the decoherence times of flux qubits, reaching up to 50 µs in recent capacitively shunted flux qubits [60].

5The Josephson energy of the smaller junction is ≈ 0.7 that of the larger junctions. 32 Chapter 2. Superconducting qubits and circuit QED

Φext

FIGURE 2.15: Fluxonium circuit.

2.6.4 Fluxonium

The fluxonium qubit has essentially the same Hamiltonian as the rf-SQUID Eq. 2.80 and effectively the same circuit. We can thus include it in the family of flux qubits. There is a fundamental difference though, that is that the inductance is not a simple geometrical inductance, but a superinductance obtained from Josephson arrays. A superinductance is not just a large inductance. It is a large inductance which does not induce a too large capacitance in parallel to it. In a geometrical realization of an inductance L this parasistic capacitance C is inevitable and it comes from the stray capacitances between the wires of the inductor, limiting the value of its characteristic impedance. As argued in Ref. [62], with geometrical inductors it is generally √ difficult to achieve characteristic impedances L/C that are larger than the vacuum impedance which is ≈ 377 Ω. In particular, it seems impossible to reduce the vacuum fluctuations of the charge variable arbitrarily (see Eq. 2.11b). A superinductance is defined as an inductance whose characteristic impedance is larger than the 2 quantum of resistance RQ = h/(2e) = 6.5 kΩ. In the fluxonium qubit the superinductance is implemented using a long array of large Josephson junctions [61, 62]. The smaller junction is usually called the black sheep junction and in normal fluxonium is not capacitively shunted. Its size is ≈ 0.05 − 0.1 that of the larger junctions in the array, and the Josephson energies scale accordingly. Fluxonium qubits have the largest relaxation times approaching 1 ms at ϕext = π. This high value of relaxation time is mainly due to suppression of the relaxation associated with quasiparticles [63,1]. As for the flux qubit there are many alternative designs of fluxonium qubits which involve capacitive shunting [64] and the use of smaller arrays of junctions [65]. In general, compared to the transmon and standard flux qubits, the fluxonium presents a richer energy level and transition structure, which can be used to design new effects that were inaccessible with previous qubits [66].

2.7 Circuit QED: artiﬁcial atoms coupled to microwave cavities

The field of cavity QED studies the light-matter interaction between an atom and the confined electromagnetic field of an optical cavity [20]. In analogy, circuit QED deals with the coupling of artificial atoms, ı.e., superconducting qubits, with microwave cavities [13, 67]. As we have already mentioned the use of microwave cavities is due 2.7. Circuit QED: artificial atoms coupled to microwave cavities 33

FIGURE 2.16: Transmon coupled to a LC oscillator. to the fact that typical plasma frequencies of Josephson junctions fall in this domain, and accordingly also typical frequencies of superconducting qubits. In Fig. 2.16 we see an example of a transmon qubit capacitively coupled to a lumped LC oscillator. We point out that the resonator is not usually made out of lumped elements, but it is realized with transmission line resonators (see Appendix A) or 3D microwave cavities [48]. Proceeding with the previously described circuit quantization procedure and projecting onto the ﬁrst two levels 6, we obtain the standard Rabi model [68] describing a two-level atom coupled to a cavity mode

Ω H = h¯ σz + ω a†a + hg¯ (a† + a)σx, (2.83) RABI 2 r where Ω and ωr are respectively the qubit and resonator frequency and the last term represents the coupling Hamiltonian. The coupling parameter g is usually much smaller than the frequency of the resonator and that of the qubit. This is always the case for light-matter interaction, and also in most superconducting qubits. However, there are proposals with superconducting qubits for obtaining a coupling parameter g that is comparable with the typical frequencies of the problem [69]. This regime is called ultrastrong coupling regime. As we said in most cases of interest this is not the case and g Ω, ωr. In this case we can make the so-called rotating wave approximation (RWA), which consists in neglegting terms σ+a† and its hermitian conjugate, which are off-resonant or equivalently are non-energy-preserving. Within this approximation we obtain the widely celebrated Jaynes-Cummings Hamiltonian [70, 20, 28] Ω H = h¯ σz + ω a†a + hg¯ (σ−a† + h.c.). (2.84) JC 2 r Superconducting qubits are capable of achieving the strong coupling regime of the Jaynes-Cummings model in which the parameter g is much larger than the typical decay rate of the cavity and of the qubit. This essentially means that we can observe the coherent effect of the coupling Hamiltonian. The Jaynes-Cummings Hamilto- nian and its generalization to many qubits (or many levels) can be used for different purposes, such as qubit readout, or to mediate the coupling between qubits. In particular, the implementation of most quantum gates is based directly or indirectly on Hamiltonians like the one in Eq. 2.84[71, 72, 73, 74]. The addition of microwave

6Being the transmon slightly anharmonic one should consider also other levels for a proper description [51]. 34 Chapter 2. Superconducting qubits and circuit QED drives is a further knob that can be used to turn on and off interactions by driving the desired transition frequencies and engineer new kinds of interactions. 35

Chapter 3

Direct parity measurement

This chapter contains material by the author that has been published in Refs. [75] and [76].

3.1 Stabilizer measurements in a nutshell

From the outset of the field of quantum computation at the beginning of the 1990s it was soon realized that, due to the intrinsic fragility of the quantum state with respect to uncontrollable environmental noise sources, some sort of quantum error correction would be necessary to perform a quantum computation. The problem of performing quantum error correction seemed almost impossible at the beginning, since quantum computation shares typical bad features of analog classical computation as the amplitudes are continuous complex variables. Additionally, one has to consider the further complications that quantum states cannot be cloned, a result that goes under the name of No-Cloning Theorem [4], and that the measurement of a quantum state destroys the quantum state itself. However, we now know that quantum error correction is possible in spite of these drawbacks. Historically, a series of quantum codes started to emerge at the beginning of the nineties culminated with Shor’s nine qubit code [77], which is obtained by concate- nating a three-qubit bit flip code with a three-qubit phase flip code. Shor’s code was the first example of a quantum code able to correct any single-qubit error. How- ever, the first QEC codes were generated by the intuition of the single authors and they were lacking a general classification. A first step in this direction was made by Calderbank and Shor [78], and independently by Steane [79], who created a class of quantum codes, which is now known as CSS codes, named after the authors. In CSS codes a QEC code is constructed starting from two classical linear codes with some specific properties. In practice, if the two classical codes are able to correct errors on t classical bits, the resulting quantum code is also able to correct arbitrary errors on t qubits. Subsequently, the theory of QEC was further refined by the introduction of the so-called stabilizer formalism [80,4]. Stabilizer codes include a broad class of QEC codes, and basically all the most popular ones, including CSS codes. The toric code [81] and its open boundary version, the surface code [82, 83], are examples of stabilizer codes, and actually also of CSS codes. In particular, these kinds 36 Chapter 3. Direct parity measurement of codes have several attractive features, above all their relatively low error threshold compared to other quantum codes, necessary to perform fault-tolerant quantum computation. For this reason, at the time of this writing, they are considered the way to go for building a fault-tolerant quantum computer [84]. Many research groups in the world, especially in the field of superconducting qubits, have undertaken the challenge to build the first logical qubit encoded using the surface code. At the time of this writing, this hasn’t been achieved so far, but we can say, without making a too bold prediction, that we are almost there. We now review some basics of the stabilizer formalism, and in particular we will later focus on a specific critical aspect, ı.e., stabilizer measurements. For a more thorough analysis of the stabilizer formalism we direct the reader to the original PhD thesis of Gottesman [80], Chapter 10 of the book of Nielsen and Chuang [4] and the review article of Terhal [85]. In what follows we denote simply by H = C2 the Hilbert space of one qubit. The

Pauli group of one qubit P1 is deﬁned as the group of all single-qubit Pauli matrices with multiplicative factors ±1, ±i:

P1 ≡ {±I, ±iI, ±X, ±iX, ±Y, ±iY, ±Z, ±iZ}, (3.1) where I denotes the 2x2 identity matrix and the Pauli matrices are deﬁned as " # " # " # 0 1 0 −i 1 0 σx ≡ X = , σy ≡ Y = , σz ≡ Z = (3.2) 1 0 i 0 0 −1

The multiplicative factors are needed to ensure that all the necessary group properties are satisﬁed. The Pauli matrices satisfy the following fundamental commutation relations [X, Y] = 2iZ , [Y, Z] = 2iX , [Z, X] = 2iY (3.3)

The Pauli group on n qubits Pn is a natural generalization as the group of all possible n-fold tensor products of Pauli matrices with prefactors ±1, ±i. For example in the case of three qubits an element of Pn is X1Z2X3. We will generically refer to elements of Pn as Pauli operators. We deﬁne the weight of a Pauli operator to be the number of locations in which it differs from the identity. We immediately point out two fundamental properties of any element of the Pauli group Pn:

• They either commute or anticommute, which readily follows from the commutation relations for single qubits Eq. 3.3. There is a simple rule for checking whether two Pauli operators commute or not. If two Pauli operators differ non-trivially in even positions, they commute, otherwise they anticommute 1.

1Non-trivially here means that we do not count differences involving the identity. For example X1 I2X3 and X1Y2X3 differ only trivially in the second position, and in fact they commute. 3.1. Stabilizer measurements in a nutshell 37

• They all have, with the exception of the identity, two eigenvalues with equal

degeneracy. This means that each element of Pn not proportional to the identity divides the total Hilbert space of dimension 2n into two eigensubspaces of equal dimension 2n/2 = 2n−1.

A stabilizer group S is an Abelian subgroup of Pn such that −I ∈/ S. We denote by S1,..., Sm the m independent generators of S and we write S = hS1,..., Smi, ⊗n meaning that S is the group generated by S1,..., Sm. We say that a ket |Ψi ∈ H is stabilized by the stabilizer group S if Sk |Ψi = +1 |Ψi ∀k ∈ {1, 2, . . . , m}. Since |Ψi is eigenket with eigenvalue +1 of all the generators of S, the same property holds ⊗n for any element of S. We denote CS ⊆ H the subspace stabilized by S, which is the code subspace. It can be shown that the dimension of the code space CS is connected with the number of generators m of S. In particular, a code space CS involving n qubits and stabilized by a stabilizer S with m generators has dimension 2n/2m = 2n−m (see Proposition 10.5 in [4]). There is a simple intuitive explanation of why this is the case and it is related to one of the two properties of Pauli matrices we highlighted before, namely the fact that each Pauli matrix divides the total Hilbert space into two eigensubspaces of same dimension. Basically, each independent stabilizer generator has eigenvalue +1 and −1, with the related +1 eigensubspace of dimension 2n/2. In practice each stabilizer generator cuts the Hilbert space in halves and so we intuitively understand that the dimension of the stabilized subspace must be 2n/2m = 2n−m. The power of the stabilizer formalism is that it gives a quite compact and clean description of many operations on the encoded qubits, and also renders the QEC cycle rather neat. In particular, a generic stabilizer code CS in active quantum error correction will be implemented by the following QEC algorithm 2. The ﬁrst basic step is

1. Encoding: the desired quantum state is encoded in the code subspace CS .

After encoding has been performed the following cycle is repeated to preserve the quantum state:

2. Errors due to environmental noise build up;

3. The stabilizer generators S1,..., Sm are measured. The result of the measurement is a vector of length m of +1 or −1 which is called the the error syndrome 3;

4. Decoding: the syndrome is processed by a classical algorithm, which deter- mines based on a certain error model, which kinds of error occurred;

5. Recovery: depending on the output of the decoding, a recovery operation is applied to the system which should correct the errors.

2Here we are just considering the implementation as a quantum memory. 3We now clearly see why S needs to be Abelian since its generators must be compatible observables. 38 Chapter 3. Direct parity measurement

Obviously, errors might also come up during the measurement, decoding and recovery operations. A great deal of effort is thus to understand to which amount errors in these operations can be tolerated so that the error correction procedure is better than the do-nothing solution, ı.e., it gives a coherence time of the encoded qubit that is longer than that of the individual physical qubits. This intuitively tells us that there should be an error threshold on each single operation that can be tolerated. Below this value the QEC procedure is effective in reducing the global error. We will not enter into the theory of fault tolerance and how to determine this threshold value, but the take home message is that each operation should introduce as low error as possible in the system and errors should not propagate [4]. In the remaining part of this chapter we will focus on step 3 in which stabilizer measurements are performed. Measurements of the stabilizer generators are also referred as parity measurements, since most of the times their result can be interpreted as the parity of a subset of qubits, either in the computational basis {|0i , |1i} or in the dual basis {|+i , |−i} (for each qubit). This is for instance the case if each stabilizer is either a product of Z operators or a product of X operators, like for the toric and surface code. We further point out that parity measurements are not only of interest for quantum error correction, but also in the context of generation of entanglement via measurement [86]. We can understand this with a simple example. Let us suppose that we have a system of two qubits initially in the product state

|0i + |1i |0i + |1i |Ψi = |+i ⊗ |+i = √ ⊗ √ . (3.4) in 2 2

By measuring the parity operator ZZ we get with probability 1/2 one of the two Bell states |00i + |11i |01i + |10i |β00i = √ , |β01i = √ , (3.5) 2 2 which are the most basic examples of entangled states. Thus, an entangled state can be generated from a product state if we have the ability to perform a parity measurement.

3.1.1 Ancilla-based parity measurement

In this subsection we explain how a general stabilizer measurement can be implemented with the help of an ancilla qubit and a quantum circuit. This is depicted in Fig. 3.1. In particular, Fig. 3.1a shows how a general measurement of a Pauli operator P can be performed with the help of an ancilla qubit prepared in the |+i state, given the ability to perform a controlled-P gate between the ancilla and the data qubits. In this way the data qubit stores only the bit of information about the two possible eigenvalues of the Pauli operator P. This information is then read out. It is very important that only this bit of information is acquired so that we do not 3.1. Stabilizer measurements in a nutshell 39

Z1Z2Z3Z4 . . . P .

|+i H |0i

(A) Circuit for measurement of a (B) Circuit for measurement of generic Pauli operator P. Z1Z2Z3Z4.

FIGURE 3.1: Ancilla-based stabilizer measurements. perform any other measurement on the state of the data qubit apart from the desired measurement of P. We point out that this is a purely quantum feature, since in the classical setting acquiring more information than necessary is not problematic. In Fig. 3.1b we see a quantum circuit implementing the measurement of the Pauli operator Z1Z2Z3Z4, which is one of the two kinds of parity measurements needed 4 for the toric code, the other being of X1X2X3X4 type . In the current designs of surface code architectures this is the way these kinds of measurement are implemented. We see that the measurement routine involves a series of 4 CNOTs with the single data qubits as control and the ancilla as a target, with the ﬁnal measurement of the ancilla that reveals the eigenvalue of the stabilizer operator. In a superconducting qubit architecture measurement of the ancilla is performed by standard single-qubit dispersive readout by coupling the ancilla qubit to a microwave resonator, and ex- ploiting the features of the dispersive Jaynes-Cummings Hamiltonian [13]. We will come back to this kind of readout in the following sections of this chapter. It is quite clear that this kind of implementation, although quite simple to understand, has some disadvantages, namely:

• it requires the presence of an ancilla and its dedicated readout;

• it requires the ability to perform CNOT gates between the data qubit and the ancilla;

• each elementary operation, ı.e., the CNOT and measurement of the ancilla might be faulty.

These reasons justify the search for alternative and more direct ways of performing a stabilizer measurement.

3.1.2 Alternative proposals for parity measurements

The search for alternative ways of performing parity measurements is certainly an active ﬁeld of research. The literature has initially focused on two-qubit parity measurements [87, 88, 89, 90, 91, 92], which is the easiest to think about. Other authors

4 Measurement of X1X2X3X4 can always be reduced to a measurement of Z1Z2Z3Z4 by applying single-qubit gates before and after the measurement. 40 Chapter 3. Direct parity measurement attempted to extend the weight of the parity measurement to three [17, 93] or four [94, 95, 96], both necessary for the surface code architecture. In [97] and in the related work [98] the authors attempted to contruct a general scheme for arbitrary parity measurement. These schemes share some common features. They are all based on the multi-qubit dispersive Jaynes-Cummings interaction between qubits and resonator, that we will discuss in details in the next section, and that has the general form (¯h = 1) N N Ωk z z † Hd = ∑ σk + ωr + ∑ χkσk a a. (3.6) k=1 2 k=1 The main feature of this Hamiltonian is that depending on the state of the qubits the resonator acquires a certain frequency. In principle, by probing this frequency the state of the data qubits can be read out. However, this is definitely not what we want since we would like to extract only one bit of information, ı.e., the one associated with the eigenvalues of the stabilizer operator we want to measure. Thus, all schemes based on this kind of interaction will have a problem of excess of information, and the main differences among the above-mentioned schemes is how this unwanted information is erased. An additional problem that all schemes have to cope with is that of unwanted intra-parity subspace dephasing. This can also be qualitatively understood in the following way. In the general setting, we have a probe that carries the information about the state of the qubits. In basically all cases this probe is the output field emitted by a cavity (or more cavities) in a scattering experiment. The state of the output system depends on the state of the qubits, and there might be an observable, a certain quadrature or the number of photons of the ouput field, that depends only on the parity, and thus reveals only the eigenvalue of the desired multi-qubit operator when measured. However, this is not sufficient for guaranteeing a good parity measurement. In fact, we need to require that the entire output state depends only on the parity. If this is not the case, the output state would carry additional information about the state of the qubits than only the parity. This information although not read out by our measurement will eventually be read out by the environment causing dephasing. We point out that the standard ancilla-based parity measurement is not exonerated by this kind of problem, which might kick in in a slightly different way. Consider for instance Fig. 3.1b. After each CNOT the ancilla acquires gradually information about the parity, but it temporarily stores more information than necessary. For instance after the first CNOT it carries information about the first qubit. This implies that if the environment probes the ancilla between two CNOT gates it will acquire more information than the only parity, causing dephasing of the data qubits. Another problem is that it is generally difficult to design schemes that are straightforwardly generalizable to parity measurements of arbitrary weight. As we said an attempt was made in Ref. [97], but the resulting scheme cannot be really considered a direct measurement since some intermediate steps are required. 3.2. Dispersive readout of superconducting qubits 41

3.2 Dispersive readout of superconducting qubits

In this section we start to familiarize with the standard dispersive readout of a single qubit. For a more detailed analysis of the single-qubit dispersive readout we refer to the original works [99, 100, 101]. This kind of readout is based on the Hamiltonian in Eq. 3.7 for a single qubit and a bosonic mode

Ω H = σz + ω + χσz a†a, (3.7) 2 r in which Ω and ωr are respectively the qubit and resonator frequency, while χ is the dispersive shift. Notice that we assume the following convention for the σz operator

σz = |ei he| − |gi hg| = |0i h0| − |1i h1| , (3.8) and we will keep it throughout this thesis unless differently speciﬁed. The Hamiltonian in Eq. 3.7 emerges naturally in the so-called dispersive regime of the Jaynes-Cummings model [102], but for illustration purposes we do not specify how it is obtained and just assume it as a model Hamiltonian 5. Hamiltonian Eq. 3.7 has a simple interpretation. If the qubit is in the excited state the cavity frequency is

ωe = ωr + χ, (3.9) while if it is in the ground state is

ωg = ωr − χ. (3.10)

By probing the cavity frequency we can thus determine the state of the qubit. This can be done by coupling capacitively (and weakly) the resonator to a transmission line. A microwave pulse at a certain carrier frequency is sent into the cavity and then reflected. The reflected signal carries the information about the state of the qubit, in particular in one of its quadrature. The quadrature with the largest amount of information about the state of the qubit is then read out by homodyne detection. We now analyze mathematically this qualitative description. We model the interaction between the resonator and the transmission line using standard input-output theory as described in Refs. [29, 103, 104]. The annihilation operator in the Heisenberg picture undergoes the following quantum Langevin equation da(t) κ √ κ √ = −i[a(t), H] − a(t) − κb (t) = −i ω + χσz(t) a(t) − a(t) − κb (t), dt 2 in r 2 in (3.11) where κ is the photon decay rate, and bin(t) is the input field. Additionally, bin(t) 0 0 satisfies the commutation relation [bin(t), bin(t )] = δ(t − t ). The output field bout(t)

5In the dispersive Jaynes-Cummings model the bare qubit frequency would also be shifted by +χ/2, which just redeﬁnes the frequency. 42 Chapter 3. Direct parity measurement

is connected to the input ﬁeld bin(t) and to the cavity ﬁeld a(t) by the input-output relation √ bout(t) = bin(t) + κa(t). (3.12)

In what follows we assume that the qubit ideally does not undergo any dissipation process. Within this assumption the qubit operator σz(t) is indeed time independent, ı.e., σz(t) = σz. Also, we assume that the qubit is initially either in the excited state |ei or in the ground state |gi. Thus, the cavity annihilation operator a(t) will have a different evolution according to the state of the qubit. We denote these two different evolutions as ae(t) and ag(t), and the same for the output ﬁeld. We can gain more insights into the solution of Eq. 3.11 by Fourier transforming. We deﬁne the Fourier transform of a generic operator A(t) in the Heisenberg picture as

1 Z +∞ A[ω] = √ dteiωt A(t). (3.13) 2π −∞

Applying the Fourier transform to Eq. 3.11, and assuming that the qubit is in the excited state we get

κ √ − iωa [ω] = −iω a [ω] − a [ω] − κb [ω], (3.14) e e e 2 e in from which we get √ κ ae[ω] = − bin[ω], (3.15) i(ωe − ω) + κ/2 and inserting into the Fourier transformed input-output relation Eq. 3.12 we get

bout,e[ω] = re(ω)bin[ω], (3.16) where we defined the reflection coefficient in the excited state

κ/2 + i(ωe − ω) re(ω) = . (3.17) κ/2 − i(ωe − ω)

Analogously, assuming the qubit in the ground state |gi we would get the reflection coefficient in the ground state rg(ω), which has the same form as Eq. 3.17 with ωe replaced by ωg. The reflection coefficient has the form (C − iDe,g)/(C + iDe,g) and has modulus one, with C = κ/2 and De,g = ωe,g − ω, as we expect since there is no internal dissipation in the cavity-qubit system. Consequently, the reflection coefficient can be expressed as

iθe,g(ω) re,g(ω) = e , (3.18) with the phase 2C D ( ) = − e,g θe,g ω arctan 2 2 . (3.19) Ce,g − D 3.2. Dispersive readout of superconducting qubits 43

0 θe

-1 θg

-2

-3 -3 -2 -1 0 1 2 3

Δd/κ

FIGURE 3.2: Phase of the reﬂection coefﬁcient as a function of ∆d = ω − ωr for the qubit in the excited (red line) and in the ground (blue line) state. In the plot χ = κ/2.

0.1

0.0

-0.1

-0.2 2 π α X -0.3 e α -0.4 g

-0.5

-0.6 -0.4 -0.2 0.0 0.2 0.4

X 0

FIGURE 3.3: Phase space evolution of the cavity ﬁeld depending on the state of the qubit. This plot was obtained using the drive amplitude shown in Fig. 3.4 and χ = κ/2.

Using the formula 1/2 arctan(2x/(1 + x2)) = arctan x, we finally express the phase of the reflection coefficient as

2(ω − ω ) χ θ (ω) = −2 arctan r ∓ . (3.20) e,g κ κ

We plot this function in Fig. 3.2, where we see clearly that we would like the average of the input ﬁeld in frequency space to be centered around the resonator frequency in order to have a high distinguishability of the qubit states. In Fig. 3.3 instead we plot the evolution in phase space of the intracavity ﬁelds

αe, αg in an interaction picture at the resonator frequency ωr. The arrows identify the direction of time. In particular, in Fig. 3.3 we assumed the average of bin(t) to be 44 Chapter 3. Direct parity measurement

0.5

0.4

0.3

κ ϵ ( t ) 0.2

0.1

0.0 0 5 10 15 20 25 κt

FIGURE 3.4: Drive amplitude used to obtain Fig. 3.3. Taking a total measurement time√ of τ = 28/κ we set the following parameters ess = 0.5 κ, σ = 4/28τ, ton = 1/28τ, toff = 16/28τ. purely imaginary and of the form

 0 , t < t ,  on   ess π  1 − cos (t − ton) , ton ≤ t < ton + σ,  2 σ  hβin(t)i = βin(t) = ie(t) = i ess , ton + σ ≤ t < toff,    ess 1 + cos π (t − t ) , t ≤ t < t + σ,  2 σ off off off   0 , t ≥ toff + σ, (3.21) with ess, ton, toff, σ real parameters. This kind of drive models a realistic ramp up and ramp down of the measurement tone. In Fig. 3.3 we have introduced the cavity quadrature operators X0, Xπ/2. We deﬁne the generic quadrature

ae−iφ + a†eiφ X = . (3.22) φ 2

With this deﬁnition we see that the averages of X0 and Xπ/2 correspond to the real and imaginary part of the average of a respectively. From Fig. 3.3 we see that the largest information about the qubit state is clearly encoded in the quadrature X0, while the orthogonal quadrature Xπ/2 does not contain any information about the state of the qubit. This is actually a consequence of the fact that we assumed the

βin to be purely imaginary. The roles between the quadratures would have been inverted if the βin were purely real. Considering the general solution of Eq. 3.11 which in the case σz(t) is independent of time as we assumed so far reads

√ Z t −iχσzt − κ t 0 −iχσz(t−t0) − κ (t−t0) 0 a(t) = a(0)e e 2 − κ dt e e 2 bin(t ). (3.23) 0 one can show indeed that if the average of bin(t) is purely imaginary then the average of Xπ/2 is independent of the state of the qubit, while the average of X0 has 3.2. Dispersive readout of superconducting qubits 45 same modulus but opposite sign depending on the state of the qubit at any time. These two features can be captured in Fig. 3.3. Additionally, also the average pho- 2 2 ton number does not depend on the state of the qubit, ı.e., |αe(t)| = |αg(t)| . From this discussion we understand that the state of the qubit can be readout by measuring the quadrature that maximizes the information content about the qubit’s state, which in our example is X0.

3.2.1 Signal-to-Noise Ratio and measurement rates

A generic quadrature of the output field can be measured by standard homodyne detection. In this kind of measurement the output field is mixed in a 50 : 50 beam splitter with a strong oscillator, that’s to say an input state at the second port that is a coherent state centered at the carrier frequency of the output field, with photon number much larger than that of the output field. The strong oscillator is usually called local oscillator and its phase selects the quadrature that is measured. In bal- anced homodyne detection the signal is proportional to the difference of the two photocurrents measured at the output ports of the beam splitter. For details about the theoretical treatment of homodyne detection see Refs. [103, 29]. In what follows we introduce the notion of measurement rate and Signal-to- Noise Ratio (SNR) similarly to Ref. [16]. The quantity that is used to determine the state of the qubit is the integrated homodyne signal whose associated operator is Z τ −iφL ∗ iφL I(τ) = dt bout(t)e + bout(t)e . (3.24) 0 Assuming that the qubit is either in the excited or ground state its average is

Z τ −iφL ∗ iφL Iσ(τ) = dt βout,σ(t)e + βout,σ(t)e . (3.25) 0 where σ = {e, g}, βout,σ(t) is the average of the output ﬁeld bout(t) depending on the state qubit, and φL the phase of the local oscillator that selects the measured quadrature 6. In the limit of strong oscillator, the integrated signal, at each time, is Gaussian distributed (see for instance Eq. 4.75 in Ref. [103]). In particular, assuming that in the initial state the qubit is either in the excited or in the ground state, we have a conditioned Gaussian distribution with average Iσ(τ) and variance equal to τ2 1 [I − I (τ)] p(I|σ)(τ) = √ exp − σ , (3.26) 2πτ 2τ2 with τ the measurement time. What we would like to know instead is the probability that given an observed signal I the qubit is in either the excited or in the ground state, ı.e., p(σ|I). Let us consider the case in which before the measurement we have complete ignorance about the state of the qubit. This amounts to saying that at time

6 Since what matters is only the difference between the two signals Ie and Ig, sometimes in the literature the contribution of βin is removed from the deﬁnition of the integrated measurement signal, as it is common to both of them. 46 Chapter 3. Direct parity measurement

t = 0 the density matrix of the system is ρ0 = 1/2(|ei he| + |gi hg|). By observing the measurement signal we now have to infer the state of the qubit. Using Bayes’ theorem we get p(σ)p(I|σ) p(σ|I) = , (3.27) p(I) where p(σ) = 1/2 from the initial density matrix and p(I) is given by the sum of the probabilities of mutually exclusive events

p(I) = p(I|e)p(e) + p(I|g)p(g). (3.28)

Since p(e) = p(g) = p(σ) we can write

p(I|σ) p(σ|I) = (3.29) p(I|e) + p(I|g)

We deﬁne the information gain about the state of the qubit given a certain observation I as the difference between the initial Shannon entropy and the the ﬁnal, conditional Shannon entropy, ı.e., the information loss of the qubit

1 I (I) = log + p(e|I) log [p(e|I)] + p(g|I) log [p(g|I)], (3.30) σ 2 2 2 2 and the average information gain is then

Z +∞ I σ = dIp(I)Iσ(I). (3.31) −∞

The measurement rate is naturally deﬁned as

dI Γ = σ . (3.32) m dτ

Γn has the units of number of bits per unit of time acquired by our measurement. The state of the qubit is determined when the information gain is equal to 1 bit. Another parameter that is usually used to characterize a measurement is the Signal-to-Noise Ratio. The SNR is easier to compute compared to the information gain and in Ref. [16] it is shown how it can be connected, within proper approximations, directly to the measurement rate. The SNR for dispersive measurements is defined as the modulus squared of the difference between the average signals associated with the excited or ground state of the qubit divided by the sum of the variances |I (τ) − I (τ)|2 |I (τ) − I (τ)|2 ( ) = e g = e g SNR τ 2 2 2 . (3.33) ∆Ie + ∆Ig 2τ The SNR intuitively quantifies the distinguishability of our signal and it is a widely used figure of merit. However, we will see that for the direct parity measurement the SNR can only be defined approximately and a treatment in terms of rigorous information gain and rates is more appropriate. 3.3. Direct dispersive parity measurement 47

3.3 Direct dispersive parity measurement

In this section we analyze the ideal model for obtaining a three-qubit parity measurement using a multi-qubit dispersive readout.

3.3.1 Concept

The ﬁrst proposal for realizing a direct two-qubit parity measurement using the dispersive readout technique was suggested by Lalumière et.al. in Ref. [87] and later implemented in [88]. The model Hamiltonian consists of two qubits dispersively coupled to one resonator mode

Ω Ω H = 1 σz + 2 σz + ω + χ σz + χ σz a†a. (3.34) 2 1 2 2 r 1 1 2 2

A necessary condition for the scheme to work is that the dispersive shifts are equal in modulus. The idea is now to look for a particular carrier frequency of the measurement tone such that the output field depends only on the parity of the two qubits and not on the particular state. As it was shown in [87] for the case of two-qubit parity measurements one can find a frequency ωd such that one of the quadratures at the steady state depends only on the parity. This quadrature is the one that is measured. However, the orthogonal quadrature would still show a dependence on the particular state of the system. Although this quadrature is not measured, the information contained in it about the state of the qubits leaks out in the environment causing additional dephasing between the parity subspaces. In order to cope with this effect it is required to work in the large |χ1,2/κ| limit. The three-qubit parity measurement scheme as first analyzed in [17] is based on a similar idea. The goal is to design the dispersive shifts and the frequencies in the problem in such a way that there exists a measurement frequency such that the output field depends only on the parity of the three qubits. From a microwave analysis of the reflection coefficient it was understood in Ref. [17] that more than one resonator was indeed needed in order to perform parity measurements with weight larger than 2. In particular, for three-qubit parity measurements two resonators are needed.

3.3.2 Conditions for three-qubit parity measurement

We will now describe in detail this scheme and we will obtain the condition that has to be matched in order to perform a parity measurement of three qubits. We will do this using directly the quantum input-output theory already used for the analysis of the single-qubit dispersive readout. We start by considering a system of three qubits and two resonators described by the following model Hamiltonian

3 2 3 3 Ωl z z † z † † H = ∑ σl + ∑ ωk + ∑ χkσl ak ak + χ12 ∑ σl (a1a2 + a1a2). (3.35) l=1 2 k=1 l=1 l=1 48 Chapter 3. Direct parity measurement

We see that we have already assumed that the dispersive shifts of the qubits for each resonator are equal. However, we didn’t assume necessarily that the dispersive shifts of different resonators are equal. We see that in addition to the standard dispersive shifts, we have included an additional term, which we interpret as a qubit state-dependent coupling of the two harmonic modes. This term is known in the literature under the name of quantum switch [105, 106]. The case analyzed in [17, 93] considered the ideal case in which χ12 = 0. However, we will show that if Eq. 3.35 is obtained perturbatively starting from the three-qubit, two-resonators Jaynes-Cummings model this term is for standard qubits non-negligible and actually complicates the parity measurement. Additionally, we have also required that each qubit gives the same quantum switch coupling coefﬁcient. These conditions guarantee already a collective property. In fact, the evolution of the system does not depend on the particular state of the qubits but only on the Hamming weight hw of the state. In this thesis we deﬁne the Hamming weight as the number of qubits in the excited state. We now consider a scattering experiment in which both resonators are coupled to a common transmission line. In this case the annihilation operators a1 and a2 undergo the coupled quantum Langevin equations

3 3 √ da1 z z κ1 κ1κ2 √ = −i ω1 + χ1 ∑ σl a1(t) − iχ12 ∑ σl a2(t) − a1(t) − a2(t) − κ1bin(t) dt l=1 l=1 2 2 (3.36a) 3 3 √ da2 z z κ2 κ1κ2 √ = −i ω2 + χ2 ∑ σl a2(t) − iχ12 ∑ σl a1(t) − a2(t) − a1(t) − κ2bin(t), dt l=1 l=1 2 2 (3.36b) with input-output boundary condition √ √ bout(t) = bin(t) + κ1a1(t) + κ2a2(t) (3.37) √ Notice that the presence of the coupling terms proportional to κ1κ2 is due to the fact that the harmonic modes are coupled to the same bath. Moreover like in Sec. 3.2 we assumed that the σz operators do not depend on time in the Heisenberg picture. Assuming that the qubits start in one of the computational basis states, which have a well deﬁned Hamming weight we will have different evolutions of the annihilation operators depending on the Hamming weight. Hence

da1,hw κ1 = −i[ω1 + χ1(3 − 2hw)]a1,hw (t) − iχ12(3 − 2hw)a2,hw (t) − a1,hw (t)− dt √ 2 κ κ √ 1 2 a (t) − κ b (t) (3.38a) 2 2,hw 1 in 3.3. Direct dispersive parity measurement 49

da2,hw κ2 = −i[ω2 + χ2(3 − 2hw)]a2,hw (t) − iχ12(3 − 2hw)a1,hw (t) − a2,hw (t)− dt √ 2 κ κ √ 1 2 a (t) − κ b (t). (3.38b) 2 1 2 in

Taking the Fourier transform as deﬁned in 3.13 we can write these equations compactly in Fourier space as " # "√ # a1,hw [ω] −1 κ1 = A (ω) √ bin[ω], (3.39) a2,hw [ω] κ2 where we deﬁned the matrix √ " κ1 κ1κ2 # i{∆ − χ (3 − 2hw)} − −iχ (3 − 2hw) − A(ω) = d1 1 √ 2 12 2 , (3.40) κ1κ2 κ2 −iχ12(3 − 2hw) − 2 i{∆d2 − χ2(3 − 2hw)} − 2 where ∆dk = ω − ωk is the detuning between frequency of the resonator k and the drive frequency. Using the input-output relation Eq. 3.37 in Fourier space and Eq. 3.39, we can write

bout,hw [ω] = r(ω; hw)bin[ω] (3.41) with the reﬂection coefﬁcient that can still be written in the form

iDhw − Chw 2 r(ω; hw) = = 1 − , (3.42) Dhw iDhw + Chw 1 + i Chw like for the case of single-qubit dispersive readout, with √ Chw = ∆d1κ2 + ∆d2κ1 + (3 − 2hw)(κ1χ1 + κ2χ2 − 2 κ1κ2χ12), (3.43a)

2 2 Dhw = 2[∆d1∆d2 + (3 − 2hw)(∆1χ1 + ∆2χ2) + (3 − 2hw) (χ1χ2 − χ12)]. (3.43b)

The condition for obtaining a perfect parity measurement is

r(ωd; hw = 0) = r(ωd; hw = 2) = re(ωd) (3.44a)

r(ωd; hw = 1) = r(ωd; hw = 3) = ro(ωd) (3.44b)

re(ωd) 6= ro(ωd), (3.44c) √ for a certain frequency ω . We look for a solution such that ∆ = ± κ /κ ∆ and √ d d1 1 2 ∆d2 = ∓ κ2/κ1∆ in which case q q κ1 κ2 χ1 − χ2 2 2 2 D κ2 κ1 (3 − 2hw) (χ χ − χ ) − ∆ hw = 2 √ ∆ + 2 1 2 12√ . Chw κ1χ1 + κ2χ2 − 2 κ1κ2χ12 (3 − 2hw)(κ1χ1 + κ2χ2 − 2 κ1κ2χ12) (3.45) 50 Chapter 3. Direct parity measurement

√ q 2 Finally, setting ∆ = 3 χ1χ2 − χ12 we can rewrite

q q κ1 κ2 √ κ χ1 − κ χ2 q Dhw 2 1 2 = 2 3 √ χ1χ2 − χ12+ Chw κ1χ1 + κ2χ2 − 2 κ1κ2χ12 2 χ1χ2 − χ12 2 f (hw) √ , (3.46) (3 − 2hw)(κ1χ1 + κ2χ2 − 2 κ1κ2χ12) where we deﬁned the function

2 (3 − 2hw) − 3 f (hw) = , (3.47) 3 − 2hw which manifestly depends only on the parity, but gives different values between even and odd parity. Consequantly also the reﬂection coefﬁcient will have the same property. To summarize, in order for a parity measurement to be possible we require the following parity condition

r q κ1 2 ∆d1 = ωd − ω1 = ± 3 χ1χ2 − χ12, (3.48a) κ2

r q κ2 2 ∆d2 = ωd − ω2 = ∓ 3 χ1χ2 − χ12. (3.48b) κ1 Let us now comment on this condition. Remembering that a typical χ is much smaller than the typical resonator frequency, we see that the two resonators need to have frequencies quite close to each other. The measurement tone needs to have a specific frequency that is always between the two resonator frequencies. Obviously, a realistic pulse would always have a transient ramp up and ramp down time and so its spectrum will not contain only the desired frequency. This represents an im- perfection of the measurement, ı.e., during the transient the measurement reveals not only information about the parity, but also partially information about the Ham- ming weight. We can understand this from Fig. 3.5 in which we plot the average of the output field bout depending on the Hamming weight. We see that during the transient the output fields corresponding to Hamming weight of same parities are distinguishable, while at the steady state they are not 7. For the analysis of the measurement it is thus important to consider not only the ramp up, but also the ramp down transient. We additionally point out that the parity condition regards the output field and not the single cavity field whose evolution depends manifestly on the particular Hamming weight. During the measurement each cavity contains information about the Hamming weight, but the emitted information contained in the output field does not, realizing a quantum eraser condition [107]. At the end of the measurement no information is left in the cavities since they both end up in the vacuum. From the parity condition Eqs. 3.48 we see that the possibility of achieving a

7Of course being coherent states here we mean approximately distinguishable. 3.3. Direct dispersive parity measurement 51

0.1

0.0

-0.1

κ -0.2 hw=0 / =1 π / 2 -0.3 hw X hw=2 -0.4 hw=3

-0.5

-0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

X0/ κ

FIGURE 3.5: Phase space evolution of the output ﬁeld. The pulse used is the same as the one in Fig. 3.4, and the parity condition is assumed to be matched. In particular, the parameters are κ1 = κ2 = κ, χ1 = χ2 = χ = κ/2, χ12 = 0. direct three-qubit parity measurement relies on the fact that it is possible to operate 2 in the parameter regime of the model Hamiltonian in which χ1χ2 ≥ χ12. We will see in the next chapter that this is non-trivial if we want to obtain our Hamiltonian effectively from a generalized Jaynes-Cummings model.

3.3.3 Measurement rates and information gain

In this subsection, we generalize the discussion of Subsec. 3.2.1 for the single-qubit readout and introduce measurement rates for the three-qubit parity measurement we discussed so far. We know that assuming the dispersive shifts χ1 and χ2, and the quantum switch coupling χ12 to be the same for all qubits, the measurement can at most reveal information about the Hamming weight. This means that we can never distinguish the states |egei and |geei for instance. Consquently we consider a setup in which before the measurement we don’t know anything about the

Hamming weight, ı.e., we assign to each Hamming weight probability p(hw) = 1/4. The Hamming weight dependent integrated signal I(τ), deﬁned in analogy with Eq.

3.24, has a Gaussian distribution p(I|hw)(τ) at each time with average

Z τ ( ) = ( ) −iφL + ∗ ( ) iφL Ihw τ dt βout,hw t e βout,hw t e , (3.49) 0 and variance τ2. Using Bayes’ theorem we can then write the probability of having a certain Hamming weight conditioned on a certain observed integrated signal I as

p(I|h ) ( | ) = w p hw I 3 . (3.50) ∑i=0 p(I|hw = 0) 52 Chapter 3. Direct parity measurement

In analogy with the single-qubit case we can deﬁne the information gain about the Hamming weight given a certain realization of the current I as the difference between the ﬁnal, conditional Shannon entropy and the initial one

3 I ( ) = + ( = | ) ( ( = | )) = hw I log2 4 ∑ p hw i I log2 p hw i I i=0 3 2 + ∑ p(hw = i|I) log2(p(hw = i|I)), (3.51) i=0 and its average Z +∞

I hw = dIp(I)Ihw (I). (3.52) −∞ The measurement rate about the Hamming weight is naturally introduced as

dI Γ = hw . (3.53) m,hw dτ

We have so far introduced the information gain about the Hamming weight and, from what we said before, we know that our measurement reveals at most 2 bits of information. We can interpret this statement as one bit giving the information about the parity, while the other bit gives the information about the particular Ham- ming weight given a certain parity. Thus, in our parity measurement we expect that the average information gain about the Hamming weight I hw does not exceed, or slightly exceeds, 1. We can introduce similarly the rate of information gain about the parity P, which is actually what we are interested in. What we need is the conditional probability that the parity is even (e) or odd (o) given a certain realization of the signal current

I. This is simply obtained since p(P = e|I) = p(hw = 0|I) + p(hw = 2|I) and

p(P = o|I) = p(hw = 1|I) + p(hw = 3|I). The measurement gain about the parity is deﬁned as

IP(I) = log2 2 + ∑ p(P = l|I) log2 p(P = l|I) = l=e,o

1 + ∑ p(P = l|I) log2 p(P = l|I). (3.54) l=e,o

The average information gain about the parity is

Z +∞ I P = dIp(I)IP(I). (3.55) −∞ and the parity measurement rate Γm,P as the derivative of the I P with respect to the measurement time dI Γ = P , (3.56) m,P dτ which has units of "parity bits" per unit of time. 3.3. Direct dispersive parity measurement 53

(A) 0.0

-0.5 P -1.0 1 - ℐ -1.5

-2.0

0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5.

χ1/κ

(B)

FIGURE 3.6: a) Average parity information gain for a measurement time τκ = 28 as a function of χ1/κ and χ2/κ. The measured quadrature of the output ﬁeld is always chosen to be the one that maximizes the information content about the state of the system. b) Missing parity information on a semilog scale for the symmetric case χ1 = χ2 (blue solid line) and the asymmetric case in which χ2 = 0.3κ (orange dashed line). In both cases, we see a minimum of the missing information (maximum information gain) at χ1 ≈ κ/2. 54 Chapter 3. Direct parity measurement

1.0

0.8

0.6

ℐ P 0.4

ℐ hw 0.2

0.0 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. χ/κ

FIGURE 3.7: Information gain about the Hamming weight and the parity. In both cases neither relaxation nor pure dephasing of the qubits is considered. The parameters are the same as in Fig. 3.6. Note that curves lie on top of each other.

With these deﬁnitions we evaluated the parity and Hamming weight information gains as a function of the dispersive shifts χ1, χ2, and assuming χ12 = 0,

κ1 = κ2 = κ, considering the piecewise pulse deﬁned in Eq. 3.21. This is shown in Fig. 3.6 where all the information gains are evaluated at the end of the measurement. In particular, in Fig. 3.6a we see that the information gain about the parity given a ﬁxed measurement time strongly depends on the dispersive shifts. This is actually the case also for the single-qubit readout in which an optimal SNR is achieved for χ = κ/2 [100]. However, we see that we can achieve parity information gains close to 1 even in the case in which the dispersive shifts are not equal to each other. We can evaluate this better in Fig. 3.6b, in which we plot the curves for χ1 = χ2 and

χ2 = 0.3χ1. We see that in both cases the minimum of the missing parity information is reached when χ1 ≈ κ/2 in analogy with the single-qubit readout. However, the minimum missing information is obtained in the symmetric case χ1 = χ2. Ad- ditionally, we see in Fig. 3.7 that for the symmetric case the information gains about the parity and the Hamming weight perfectly overlap, which means that the measurement correctly reveals almost solely information about the parity.

3.3.4 Quantum trajectories and stochastic master equation

In order to analyze in detail the evolution of a quantum system under a continuous weak measurement, like for the case of the dispersive readout, we need to use quantum trajectory theory which is formulated in terms of stochastic master equations [103, 29, 108, 109, 110]. In the general setting of open quantum systems we have a system described by a certain Hamiltonian HS, another system that we call the environment or the bath with Hamiltonian HE, and an interaction between them

HI [111]. The whole system has Hilbert space H = HS ⊗ HE, with HS and HE the Hilbert spaces of the system and the environment respectively, and, being overall a 3.3. Direct dispersive parity measurement 55 closed system, undergoes the Schrödinger equation with Hamiltonian 8

H = HS + HE + HI. (3.57)

If the initial state is a pure state at time t0 and we can then write the initial density matrix as ρtot(t0) = |ψ(t0)i hψ(t0)|, it will then remain in a pure state at all times,

ı.e., ρtot(t) = |ψ(t)i hψ(t)|. However, our goal is not to obtain |ψ(t)i, but rather to be able to compute the average of any operator acting only on the quantum system.

This means that we are considering operators that can be written in the form O ⊗ IE, with IE the identity on the Hilbert space of the environment. It can be shown that in order to obtain the averages of the system operators it is sufﬁcient to know the so-called reduced density matrix (see Ref. [111] Eq. 2.92) of the system

ρ = TrE(ρtot) ≡ ∑ hi|E ρtot |iiE , (3.58) i where TrE denotes the partial trace on the environment and {|iiE} is a basis for the environment Hilbert space. The goal is then to obtain ρ(t) at each time so that we have complete knowledge about the system. It is useful at this point to state some fundamental properties of any density matrix ρ:

• unit trace: Tr(ρ) = 1;

• positivity: ρ ≥ 0;

• hermiticity: ρ = ρ†.

In general any transformation of a density matrix ρ0 can be written in the form of a so-called quantum operation [4]

† ρ = E(ρ0) = ∑ Eiρ0Ei , (3.59) i with the fundamental condition

† ∑ Ei Ei = I, (3.60) i with I denoting the identity of the quantum system. The operators Ei are usually † called Kraus operators, while the operators Ei Ei are known as effects. One can show that if ρ0 satisfies the properties of a density matrix then if ρ is given by a quantum operation like in Eq. 3.59 it also satisfies these properties. Eq. 3.59 is a completely general transformation of a density matrix into another well defined density matrix. This means that also the evolution in time of a density matrix ρ(t) starting from an initial density matrix ρ0 must be written at each time like in Eq. 3.59, with time- dependent Kraus operators. Thus, in principle knowledge of the quantum operation at each time identifies the quantum state at any time. However, we immediately

8We are not considering the possibility of measurement of certain observables yet. 56 Chapter 3. Direct parity measurement see that this is practically quite inconvenient and hard to do. What we would like instead is to write the evolution of a quantum system in the form of a differential equation involving ρ and its ﬁrst derivative

dρ = L(t)ρ(t), (3.61) dt with L(t) usually called Liouville superoperator 9. In addition, we would like also to treat master equations which always preserve the properties of a density matrix and that that are possibly local in time, ı.e., L is independent of time. This last property is known as markovianity, and it has to be taken as an approximation. It can be shown that the most general Markovian master equation that preserves the properties of a density matrix is of the form [111, 112]

2 dρ N −1 = LLρ(t) = −i Heff, ρ + ∑ γkD[Lk]ρ, (3.62) dt k=1 where N is the dimension of the quantum system we are considering, the coefﬁcients 10 γk ≥ 0 , and we deﬁned the dissipator

1 1 D[L ]ρ = L ρL† − L† L ρ − ρL† L . (3.63) k k k 2 k k 2 k k

Eq. 3.62 is usually known as Lindblad master equation and the operators Lk as Lind- blad operators. Notice that in Eq. 3.62 we have introduced an effective Hamiltonian

Heff, which is not necessarily the same as the bare Hamiltonian of the system. This is because in the process of the elimination of the environmental degrees of freedom, some additional terms come out that add to the bare system Hamiltonian. A promi- nent example is the Lamb shift. It is easy to show that considering an infinitesimal interval dt, ρ(t + dt) can be cast in the form of a quantum operation Eq. 3.59 if Eq. 3.62 is satisfied, and thus guarantees that all the properties of a density matrix are also automatically satisfied. Lindblad master equations are the standard method of analysis of relaxation and pure dephasing processes. Considering a qubit for in- − stance relaxation would be described by a Lindblad dissipator γ1D[σ ]ρ. This term causes an exponential decay at rate γ1of the diagonal component of the density matrix ρee = he| ρ |ei, which is just the probability of finding the system in the excited state. However, this is not the only effect since it causes also an exponential decay of the off-diagonal components of the density matrices ρeg = ρge, usually called the coherences, at a rate γ1/2. Thus, when we have relaxation we also have an associated dephasing. We use interchangeably the words dephasing and decoherence, referring to the decay of the diagonal components of the density matrix in the chosen basis, ı.e., of the coherences. We refer to pure dephasing if we have a process

9The term superoperator refers to the fact that it transforms an operator into another operator. Also a quantum operation is a superoperator. 10This is a necessary condition for the preservation of positivity. 3.3. Direct dispersive parity measurement 57

z that causes only the decay of the coherences. A Lindblad term γϕ/2D[σ ]ρ causes only pure dephasing at a rate γϕ. Deﬁning the relaxation times T1 = 1/γ1, the pure dephasing time Tϕ = 1/γϕ and the total decoherence time T2 = 1/(γ1/2 + γz) we get the well known relation 1 1 1 = + . (3.64) T2 2T1 Tϕ

T1, T2, Tϕ are common measures of the quality of a qubit, and one should keep in mind that their definition already assumes a Markovian setting. As a last example, considering a harmonic oscillator, the decay of the photon number is described by a Lindblad term κD[a]ρ. We have so far considered only the case in which the total system undergoes a unitary evolution. However, we want to analyze how a measurement influences the evolution of the quantum system, in particular considering the case of weak measurements that take place in a finite amount of time. This is in contrast with an idealized projective measurement in which the collapse of the quantum state into one of the eigenkets of the measured observable happens instantaneously. In order to extend our open quantum system analysis to the case in which also weak measurements are presents we need two things:

1. we need to retain a description of the environment that allows us to describe how the information about a certain system’s observable is transferred to the environment. This leads to the input-output theory that has been used in Sec. 3.2 and Subsec. 3.3.2;

2. we need to understand what is the backaction on the quantum state given a certain observed measurement signal. This leads to quantum trajectory theory.

We immediately point out that the notion of markovianity is usually a fundamental assumption of both input-output theory and of quantum trajectory theory. How- ever, attempts to go beyond the markovian approximation have been made both for input-output theory [113, 114] and in the theory of quantum trajectories [115, 116, 117]. In quantum trajectory theory, since the measurement of a certain observable has a random component, we expect also the associated backaction on the system to be random. Thus, on top of the deterministic evolution that can be described by a markovian master equation, or in general by a quantum operation, the measurement adds a stochastic component. We additionally expect this stochastic process to be discon- tinuous in the sense that we expect the system to undergo simply the deterministic evolution for some time, until a detection takes place. The detection events are called quantum jumps. In our problem of dispersive readout, being it single-qubit or multi- qubit, we have in general the problem of reading out the quadrature of the output ﬁeld emitted by a cavity, or more cavities, via homodyne detection that we brieﬂy discussed at the beginning of Sec. 3.2.1. We then need the quantum trajectory equation associated with homodyne detection, which in the limit of strong local oscillator 58 Chapter 3. Direct parity measurement gives rise to a continuous stochastic evolution given by a stochastic master equation (SME) which in Itô form reads (see Eq. 4.72 in Ref. [103])

dρJ(t) = LLρ(t)dt + dW(t)M[c]ρJ(t), (3.65) where LL is a generic Liouville superoperator in Lindblad form like in Eq. 3.62, while the measurement superoperator is deﬁned as

† † M[c]ρJ = cρJ + ρJc − Tr(cρJ + ρJc )ρJ, (3.66) where the system operator c is in general the contribution of the system to the output ﬁeld multiplied by exp[−iφ ], with φ the phase of the local oscillator that selects the L L √ measured quadrature. For instance in the single qubit readout c = κa exp[−iφL], while for the three-qubit parity readout with two resonators the operator c would be √ √ c = ( κ1a1 + κ2a2) exp[−iφL]. The quantity dW(t) is the Wiener increment, ı.e., a random variable that at each time is Gaussian distributed with mean and variance

E[dW(t)] = 0, (3.67a)

E[dW(t)2] = dt. (3.67b)

The presence of a continuous Wiener increment means that the system now undergoes a diffusive process rather than a jump process. Finally, the measurement signal in the continuous limit is given by

dW J (t) ≡ hx i (t) + , (3.68) hom φL J dt

† with xφL = c exp[−iφL] + c exp[iφL]. Eq. 3.68 justifies why in the analysis of the measurement rates presented in Subsecs. 3.2.1 and 3.3.3 the integrated measurement signal was assumed Gaussian distributed with variance equal to τ2, with τ the measurement time. A single trajectory generated by the SME of homodyne detection Eq. 3.65 would correspond in the physical setting to a single experimental run in which we get a certain noisy measurement signal, and the state of the quantum system gradually collapses (hopefully) in one of the eigenkets of the observable we want to measure. In addition, we identify the measurement result from the integrated signal according to a certain criterion. For instance in the single-qubit case the average signal with qubit prepared in the ground state is positive, while it is negative if prepared in the excited state. Thus, a simple criterion is to say that we measure the ground state if the integrated signal is larger than zero, while the excited state otherwise. At any rate any approach will lead to some misidentification of the quantum state. By generating numerically many trajectories using one of the numerical methods available to integrate Eq. 3.65[118], we can obtain information about the averages and statistical distributions of any quantity. For instance a quantity of interest is 3.3. Direct dispersive parity measurement 59 the average quantum state fidelity of the measurement process. The fidelity is a measure of how close two quantum states with density matrices ρ and σ are close to each other and it is defined as [4]

q F(ρ, σ) = Tr ρ1/2σρ1/2 , (3.69) and it has the property 0 ≤ F(ρ, σ) ≤ 1. In practice, after each quantum trajectory simulation, from the result of the measurement signal we infer that the system is in a certain quantum state and we compute the fidelity of the density matrix that underwent the stochastic evolution with respect to the expected state. The average of all these fidelities is the average fidelity. It is clear that errors in the identification of the quantum state from the measurement signal reduce the fidelity. For instance in the single-qubit case the state might have collapsed to the excited state, but from the measurement signal we conclude that the system is in the ground state, and computing the fidelity we would get zero.

3.3.5 Quantum trajectories for three-qubit parity measurement

In principle Eq. 3.65 can be used to analyze the measurement process in the three- qubit parity measurement we described. However, since we have to simulate two cavity modes (including at least 5 Fock states) and three qubits, we immediately end up with quite a large computation. What we are interested in is actually only the reduced master equation for the qubits and so it would be better to obtain an equivalent stochastic master equation for the qubits only. As it was shown in Ref. [100] this is possible for the single-qubit measurement, by employing a so-called polaron transformation. In Ref. [87] the same method was used for the two-qubit parity measurement. The polaron transformation approach can also be used to obtain the SME for the three-qubit parity measurement with model Hamiltonian Eq. 3.35. In Ref. [75], we also discussed an alternative, simpler derivation based on an ansatz. We consider a system of three qubits coupled dispersively to two resonators with Hamiltonian 11

3 2 3 Ωl z z † Htot(t) = ∑ σl + ∑ ∆dk + ∑ χkσl ak ak+ l=1 2 k=1 l=1 3 z † † √ † √ † χ12 ∑ σl (a1a2 + a1a2) + ed(t) κ1 a1 + a1 + κ2 a2 + a2 , (3.70) l=1

12 with ∆dk = ωk − ωd the detuning between resonator frequency and drive . In the total master equation we consider the presence of pure relaxation and dephasing processes for the qubits, as well as correlated photon decay of the resonators. We

11 We are considering an interaction picture at the drive frequency. √ 12 ( )[ ( ( ) + ) + √ The drive in the Schrödinger picture would be ed t κ1 a1 exp iωdt h.c. κ2(a2 exp(iωdt) + h.c.)]. 60 Chapter 3. Direct parity measurement then consider the following Liouville superoperator in Lindblad form

3 3 γ √ √ L = − [ ( ) ] + D[ −] + z,l D[ z] + D[ + ] Lρt i H t , ρt ∑ γ1,l σl ρt ∑ σl ρ κ1a1 κ2a2 ρ, (3.71) l=1 l=1 2 with ρt the density matrix of the total system of qubits+resonators, γ1,l the relaxation rate of qubit l, γz,l the pure dephasing rate of qubit l, and κ1, κ2 the photon decay rates of resonator 1 and 2 respectively. The starting SME we consider is Eq. 3.65 with √ √ √ Liouville operator given by Eq. 3.71 and c = η( κ1a1 + κ2a2) exp[−iφL], where we also considered the quantum efﬁciency of the detector η. The elimination of the cavity degrees, following Refs. [75, 100], gives the effective SME for the qubits

3 3 γ = − [ ] + D[ −] + ϕ,l D[ ] dρJ i HQ, ρJ dt ∑ γ1,l σl ρJdt ∑ σz,l ρJdt l=1 l=1 2 √ −iφL − ∑ (Γd,xy + iAxy)ΠxρJΠydt + ηM ∑ βout,xΠxe ρJdW, (3.72) x6=y x with the qubit Hamiltonian HQ simply given by

3 Ωl z HQ = ∑ σl . (3.73) l=1 2

In the previous formula the labels x or y denote a specific state of the three qubits. Hence, a sum on x, for instance, means a sum over all possible states of a three-qubit system, and Πx denotes a projector on that specific state, ı.e., Πx = |xi hx|. The qubit- √ √ state dependent output field is βout,x = κ1α1,x + κ2α2,x, where α1,x and α2,x are the qubit-state dependent average cavity amplitudes for cavity 1 and 2 respectively 13. In particular, they obey the following evolution equations: √ dα κ κ κ √ 1,x = −i(∆ + χ )α (t) − iχ α (t) − 1 α (t) − 1 1 α (t) − i κ ε (t), dt d1 1,x 1,x 12,x 2,x 2 1,x 2 2,x 1 d (3.74) √ dα κ κ κ √ 2,x = −i(∆ + χ )α (t) − iχ α (t) − 2 α (t) − 1 2 α (t) − i κ ε (t), dt d2 2,x 2,x 12,x 1,x 2 2,x 2 1,x 2 d (3.75) where by χ1,x, χ2,x and χ12,x, we denote respectively the frequency shift of resonator 1 and 2, and the coupling coefficient between the two resonators, in the state is |xi. The measurement-induced dephasing rates and Stark shifts are given by:

∗ ∗ Γd,xy = (χ1,y − χ1,x)Im{α1,xα1,y} + (χ2,y − χ2,x)Im{α2,xα2,y} ∗ ∗ + (χ12,y − χ12,x)[Im{α2,xα1,y} + Im{α1,xα2,y}], (3.76a)

13Formally this is not the output ﬁeld, but only the part of the output ﬁeld that depends on the system. 3.3. Direct dispersive parity measurement 61

1200

1000

800

600 Frequency 400

200

0 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 p F− = hψ−| ρ |ψ−i

FIGURE 3.8: Histogram of the ﬁdelities when the state identiﬁed from the measurement signal is |ψ−i. The histogram corresponds to 5031 trajectories. The simulations were generated taking χ to be all equal. The dephasing rate is taken to be equal for all qubits and set as γz = χ/300, while the relaxation rate was set to zero. The measurement time is taken to be τχ = 10 with a time-dependent drive as in Eq. 3.21. In addition, κ1 = κ2 = κ = 2χ.

∗ ∗ Axy = (χ1,y − χ1,x)Re{α1,xα1,y} + (χ2,y − χ2,x)Re{α2,xα2,y} ∗ ∗ + (χ12,y − χ12,x)[Re{α2,xα1,y} + Re{α1,xα2,y}]. (3.76b)

Finally, the measurement signal can be computed as √ −iφ ∗ iφ Jhom(t)dt = ηTr ∑ βout,xe + βout,xe ΠxρJ dt + dW(t). (3.77) x

|ψ+i and |ψ−i have definite parity, but do not have a definite Hamming weight. In Fig. 3.8 we plot the histogram of the fidelities obtained when the state identified 14 from the integrated measurement signal is |ψ−i . The average fidelity in this case is ≈ 94%.

14Fig. 3.8 was generated by my co-author Ben Criger who performed the simulations.

Chapter 4

Tunable Coupling Qubit for parity measurements

This chapter contains material by the author that has been published in Ref. [76].

4.1 Quantum switch term for a transmon

In the previous chapter we have shown that in order to satisfy the parity condition

Eq. 3.48, it is fundamental to be able to achieve a set of parameters for which χ1χ2 ≥

χ12. In this section we show that this condition cannot be achieved with a single transmon qubit, if we want to obtain the effective model Eq. 3.35 perturbatively from a generalized Jaynes-Cummings model. In order to see this let us consider a single transmon, that we treat as a Dufﬁng oscillator, that is bilinearly coupled to two resonators. The Hamiltonian in the RWA can be written as (¯h = 1)

2 † † H = HDuff + ∑ ωkak ak + gk(akb + h.c.) = H0 + Vint, (4.1) k=1 where gk are the linear coupling constants between transmon and resonators, and the Hamiltonian of a Dufﬁng oscillator is given in Eq. 2.79. For later convenience we have also divided the Hamiltonian in its diagonal part

2 † H0 = HDuff + ∑ ωkak ak, (4.2) k=1 and off-diagonal part, that represents the interaction

2 † Vint = ∑ gk(akb + h.c.). (4.3) k=1

We are interested in obtaining an effective dispersive Hamiltonian that describes the system when only the ground state |gi and the ﬁrst excited state |ei of the transmon are populated, and also few photons are present in the resonator. In particular, this Hamiltonian will be valid when the detuning between transmon and both resonator frequencies is large compared to the linear couplings g1,2. As argued in Ref. [51], 64 Chapter 4. Tunable Coupling Qubit for parity measurements

2ωt + δ | f i

ωt |ei

Res 1

Res 2 |gi

FIGURE 4.1: First three energy levels and transitions diagram for a transmon. The orange and green arrows denote the direct transitions that can be caused by the interaction with the resonator 1 and 2 respectively, when the transmon is coupled linearly to them. As we see the transitions are the same. the correct procedure to perform this perturbative analysis is not to project directly the Hamiltonian Eq. 4.1 onto the first two levels of the transmon, but to keep also the third level | f i, perform the dispersive transformation,and finally project onto the first two levels. This is due to the ladder-like structure of the transmon transitions, as shown in Fig. 4.1, and to the fact that the anharmonicity of the transmon is small, so that the effect of the level repulsion induced by the third level is not negligible. By projecting Eq. 4.1 onto the first three levels of the transmon we get

2 √ † H = Ωe |ei he| + Ω f | f i h f | + ∑ ωkak ak + gk(|ei hg| ak + h.c.) + 2gk(| f i hg| ak + h.c.), k=1 (4.4) with Ωe = ωt and Ω f = 2ωt + δ. Assuming that both interactions of the transmons with the resonators are in the √ dispersive regime, which mathematically means |gk/∆k| 1 and | 2gk/(∆k +

δ)| 1, with the detuning ∆k = Ωe − ωk and k = 1, 2. In this case, we can employ a Schrieffer-Wolff transformation that removes the interaction between the transmon and the resonators [119, 120]. The second order Schrieffer-Wolff transformation can † be written as U = exp[S1] with S1 a antihermitian S1 = −S1 and block-diagonal operator satisfying the condition

[S1, H0] = −Vint. (4.5)

† The effective Hamiltonian is then obtained as Heff = UHU keeping only terms up to second order in |gk/∆k| and |gk/(∆k + δ)|. Using the Baker-Campbell-Hausdorff (BCH) formula which for general operators A and B reads

+∞ A −A 1 1 e Be = B + [A, B] + [A, [A, B]] + ··· = ∑ Cn(A, B), (4.6) 2! n=0 n! 4.1. Quantum switch term for a transmon 65

with Cn(A, B) the n-th commutator between A and B

Cn(A, B) = [A,..., [A, B] ... ]] . (4.7) | {z } n times

In this way we get that we can simply express the effective Hamiltonian as

[S , V ] H = H + 1 int . (4.8) eff 0 2

This is a general result, that is not limited to the system we are considering that we will apply also in the following sections. In our case the operator S1 satisfying Eq. 4.5 is found to be √ 2 g 2g S = k |ei hg| a + k | f i he| a − h.c.. (4.9) 1 ∑ k + k k=1 ∆k ∆k δ

The associated effective Hamiltonian obtained from Eq. 4.8 reads

By projecting this Hamiltonian onto the subspace spanned by the excited and ground state we can write the dispersive Hamiltonian as

¯ 2 Ω z z z † Hd = σ + ∑ (ω¯ k + χkσ ) + (χ¯12 + χ12σ )(a1a2 + h.c.), (4.11) 2 k=1

¯ 2 2 where the effective qubit frequency is given by Ω = Ωe + ∑k=1 gk /∆k, while the 2 effective resonator frequencies are ω¯ k = ωk − gk /(∆k + δ). Additionally

2 2 gk gk χk = − , (4.12a) ∆k ∆k + δ 1 g2 g1 χ12 = χ1 + χ2 , (4.12b) 2 g1 g2 g1g2 1 1 χ¯12 = − + . (4.12c) 2 ∆1 + δ ∆2 + δ 66 Chapter 4. Tunable Coupling Qubit for parity measurements

We see that there is a functional dependence between χ12 and the dispersive shifts

χ1, χ2. In particular, it is immediate to check that the previous formulas imply χ1χ2 − 2 χ12 ≤ 0. From this we might conclude that the parity condition cannot be matched. However, this is not so straightforward. In fact, from Eq. 4.11 we see that apart from the dispersive shifts and the quantum switch term that we considered so far we also get a direct coupling between the two resonators with coupling coefficient χ¯12. We will now comment on this term and show that, indeed, it does not change our analysis. First of all, we notice that in the limit of infinite (positive) anharmonicity this term is actually zero. This is the limit in which we recover the behaviour of the simple two-level system, since all the terms that depend on the anharmonicity drop out. Thus, for a two-level system we cannot match the parity condition. In case we have finite anharmonicity like for the transmon, we can actually diagonalize the part of the Hamiltonian that involves only the resonators. A way of doing this is to perform a so-called Bogoliubov transformation. We express the annihilation operators of the resonators as " # " #" # a cos θ sin θ a˜ 1 = 1 , (4.13) a2 − sin θ cos θ a˜2

† † where a˜1, a˜1, a˜2, a˜2 obey bosonic commuation relations. It is easy to check that with † † this deﬁnition a1, a1, a2, a2 also correctly obey bosonic commuation relations. The idea is then to plug Eq. 4.13 into the Hamiltonian Eq. 4.11 and look for θ such that no † interactions of the form ∼ a˜1a˜2 + h.c. appear. Without even selecting the particular value of θ we can show immediately that this will not allow the parity measurement condition to be fulﬁlled. In fact, we would get in the new basis effective dispersive shifts and effective quantum switch couplings given by

2 2 χ1,eff = χ1 cos θ + χ2 sin θ − χ12 sin 2θ, (4.14a)

2 2 χ2,eff = χ1 sin θ + χ2 cos θ + χ12 sin 2θ, (4.14b) 1 χ = χ cos 2θ + (χ − χ ) sin 2θ. (4.14c) 12,eff 12 2 1 2 2 2 One can check that these expressions imply χ1,effχ2,eff − χ12,eff = χ1χ2 − χ12 ≤ 0, which ﬁnally tells us that also with transmon qubits the parity condition cannot be matched. In order to understand how to solve this problem, we need to understand what the origin of the quantum switch term is. We claim that one gets a quantum switch term between two resonators, after a dispersive transformation, whenever the two resonators can cause the same transitions, which is the case depicted in Fig. 4.1 for the transmon for instance. Instead, if we were able to engineer a situation in which resonator 1 can cause the transition |gi ↔ |ei, but not the transition |ei ↔ | f i, and vice versa for resonator 2, the quantum switch term would be zero. This situation is depicted for instance in Fig. 4.2 for the level structure of the transmon. In this 4.2. Tunable Coupling Qubit 67

2ωt + δ | f i

ωt |ei

Res 1

Res 2 |gi

FIGURE 4.2: Example of transition structure that would give zero quantum switch term.

case, the dispersive shifts χ1 and χ2 are still non-zero. At the current state, we do not know a way of engineering this property with transmon qubits. Standard capacitive coupling of a transmon to two resonators leads to the standard Hamiltonian analyzed in this section Eq. 4.1, which does not have the feature depicted in Fig. 4.2 as we have seen. In the following part of this chapter we will analyze a qubit that does show the basic feature we need to set the χ12 term to zero.

4.2 Tunable Coupling Qubit

In this section we describe the so-called Tunable Coupling Qubit (TCQ). We will later show how this qubit with simple capacitive coupling can break the constraint on the quantum switch term discussed in the previous section. The TCQ was first introduced in Ref. [121], and further analyzed in Refs. [122, 123]. The basic circuit of the TCQ is shown in Fig. 4.3, where for completeness we also included voltage sources, that will allow us to analyze the effect of charge noise, like for a single CPB. From the circuit we see that the TCQ is essentially a system made up of two capacitively coupled transmons. In particular, the coupling is strong meaning that the coupling capacitance is of the same order as the capacitances of the transmons. The idea is to use the TCQ to encode a single qubit in its two lowest level in order to obtain a qubit with higher flexibility compared to the simple transmon, while still retaining similar charge noise insensitivity. In particular, it has been shown how the TCQ can be operated in a way in which it is protected from the Purcell effect, while still being measurable via dispersive readout. As the name suggests the TCQ can also be used to implement a qubit whose coupling strength to the resonator can be tuned independently of its frequency. This can be done by substituting the Josephson junctions of the TCQ with SQUID loops, whose Josephson energy can be tuned by controlling the magnetic flux in the loop. This can also be done with a single transmon of course, but since by flux tuning we just modify the Josephson inductance, the frequency of the transmon and its impedance, which controls the coupling strength, cannot be tuned independently. In Ref. [124] also the possibility to suppress photon shot noise dephasing was explored. In Refs. [125, 126] a similar system was proposed employing inductive coupling instead of capacitive coupling. 68 Chapter 4. Tunable Coupling Qubit for parity measurements

Cg+ CI Cg− Φ+ Φ−

+ + Vg+ − CJ+ EJ+ CJ− EJ− − Vg−

FIGURE 4.3: Basic circuit of the TCQ. The Josephson junctions can be substituted by ﬂux tunable SQUID loops allowing for the control of the Josephson energies EJ±.

Using the standard circuit quantization procedure, we can write the Hamiltonian of the TCQ as

2 (Q± + Cg±V±) 2π HTCQ = ∑ − ∑ EJ± cos Φ± + ± 2C˜ J± ± Φ0 1 (Q+ + Cg+Vg+)(Q− + Cg−Vg−), (4.15) C˜I where Φ± and Q± are conjugate variables, which after quantization will have commutation relations [Φ+, Q+] = [Φ−, Q−] = ih¯ . We additionally deﬁned the effective capacitances 1 C + C ∓ = I Σ , (4.16a) C˜ J± CICΣ+ + CICΣ− + CΣ+CΣ− 1 C = I , (4.16b) C˜I CICΣ+ + CICΣ− + CΣ+CΣ− with CΣ± = CJ± + Cg±. Introducing the number of tunnelled Cooper pairs Q± =

2eN± and the superconducting phases ϕ± = 2πΦ±/Φ0 we can rewrite the Hamilto- nian as

2 HTCQ = ∑ 4EC±(N± − ng±) − ∑ EJ± cos ϕ± + 8EI (N+ − ng+)(N− − ng−), (4.17) ± ±

2 2 where we deﬁned the charging energies EC± = e /(2C˜ J), EI = e /(2C˜I ), and the reduced charges ng± = −Cg±Vg±/(2e). Notice that according to our deﬁnition the number and phase operators now obey the commutation relations [ϕ+, N+] =

[ϕ−, N−] = i. We know that for the transmon qubit in the limit of large EJ/EC the low lying energy levels become insensitive to charge noise, ı.e., they do not depend on the reduced gate charge and noise associated with it. This means that the system becomes insensitive to decoherence induced by charge noise [51, 23]. We expect this to be true also for the TCQ in the limit in which EJ+/EC+, EJ−/EC− 1. We con- ﬁrm this by diagonalizing numerically the Hamiltonian in the charge basis as shown in Fig. 4.4. Assuming the transmons composing the TCQ to be equal, we see that for EJ/EC the spectrum is practically ﬂat as a function of the parameters ng±. This means that we can drop these parameters from our Hamiltonian when we work in 4.2. Tunable Coupling Qubit 69

(A) (B)

FIGURE 4.4: First 6 energy levels of the TCQ Hamiltonian Eq. 4.17 as a function of the offset charges ng+ and ng− (m = {0, 1, 2, 3, 4, 5}). The zero of the energy is taken to be in each plot the minimum of the energies of the lowest level. The transmons are considered symmetric in this case and EI /EC = 1. The plots are obtained by direct numerical diagonalization of HTCQ by writing it in the charge basis and truncating the Hilbert space. this regime. Additionally, as for the transmon we will expand each Josephson potential up to fourth order in the superconducting phases. This gives the effective potential EJ± 2 EJ± 4 UJ,e f f (ϕ+, ϕ−) = ∑ −EJ± + ϕ± − ϕ±. (4.18) ± 2 24 We introduce annihilation and creation operators for the transmon modes

1/4 2EC± † ϕ± = b± + b± , (4.19a) EJ±

1/4 i EJ± † N± = b± − b± , (4.19b) 2 2EC±

† † which satisfy bosonic commuatation relations [b+, b+] = [b−, b−] = 1. Plugging Eqs. 4.19 into Eq. 4.17 and performing several rotating wave approximations, ı.e., neglecting terms with unequal number of annihilation and creation operators, we obtain the effective Hamiltonian of the TCQ in the form of two linearly coupled 70 Chapter 4. Tunable Coupling Qubit for parity measurements

Dufﬁng oscillators (¯h = 1):

RWA † δ± † † † † HTCQ = ∑ ω±b±b± + b±b±b±b± + J b+b− + b+b− , (4.20) ± 2 p with the mode frequencies ω± = 8EC±EJ± − EC±, anharmonicities δ± = −EC±, and the linear coupling parameter

1/4 1/4 EJ+ EJ− J = 2EI . (4.21) 2EC+ 2EC−

At this point the idea is to approximately diagonalize the Hamiltonian assuming weak anharmonicities. As we discussed in Sec. 4.1, we can introduce a Bogoliubov transformation for the transmon modes akin to Eq. 4.13 " # " #" # b+ cos λ sin λ b˜+ = . (4.22) b− − sin λ cos λ b˜−

Plugging this transformation into Eq. 4.17, we obtain that the parameter λ that cancels the linear interaction is given by

1 2J λ = arctan − , (4.23) 2 ξ

1 with ξ = ω+ − ω− − 2(δ+ − δ−) , and the Hamiltonian becomes

˜ ˜ RWA ˜† ˜ δ± ˜† ˜† ˜ ˜ ˜ ˜† ˜ ˜† ˜ HTCQ = ∑ ω˜ ±b±b± + b±b±b±b± + δcb+b+b−b−, (4.24) ± 2 where we obtained the characteristic frequencies of the diagonal modes of the TCQ and their anharmonicities in terms of the bare ones as

ω+ + ω− ω˜ ± = ± (ω+ − ω−) cos(2λ) ∓ J sin(2λ), (4.25a) 2

δ+ + δ− 2 δ+ − δ− δ˜± = [1 + cos (2λ)] ± cos(2λ). (4.25b) 2 2 The self-anharmonicities remain of the same order of magnitude as those of the bare transmons. This means that the TCQ cannot be considered a system that preserves charge noise insensitivity of the transmon, while increasing the anharmonicity. We see however that in this approximately diagonal basis we also have a cross-Kerr term

δ+ + δ− δ˜ = sin2(2λ). (4.26) c 2

This is an important feature of the TCQ which distinguishes it from two coupled two-level systems. In particular, its presence implies that the energy of the level

1We conventionally deﬁne the + mode to be the one for which ξ is positive. 4.2. Tunable Coupling Qubit 71

2ω ˜ + + δ˜+ |2+0−i |1+1−i ω˜ + + ω˜ − + δ˜c

2ω ˜ − + δ˜− |0+2−i ω˜ + |1+0−i

ω˜ − |0iL = |0+1−i b˜ + b˜ −

|1iL = |0+0−i

FIGURE 4.5: First six energy levels and transitions of a TCQ. The states in which the qubit is encoded are shown in red. Again we use the convention that the ﬁrst excited state is the logical zero state, while the ground the logical one state.

|1+1−i is not equal to the sum of the energies of states |1+0−i, |0+1−i. We can see this from Fig. 4.5, in which we show the ﬁrst six levels of the TCQ and also the transitions that can be caused by the bosonic operators b˜± and their hermitian conjugates. As shown in Fig. 4.5 the qubit is encoded in the ﬁrst two levels of the system.

4.2.1 Purcell protection

As we stated in the introductory part of this section, the TCQ can be used in a configuration in which it is protected from Purcell effect, but still measurable via dispersive readout by coupling to a resonator [121]. In this subsection, we will briefly explain why this is the case. This property turns out to be closely related to the fact that the TCQ can be used for direct three-qubit parity measurements as we will later show in this chapter. The Purcell effect is basically the decay induced by the coupling of the system to a leaky cavity [28]. This decay can be enhanced or suppressed compared to the case in which there is no cavity, depending on whether the system is resonant or not. Let us thus consider a TCQ, in which both bare transmons are coupled linearly to a cavity mode, with photon decay rate κ. The Hamiltonian in the bare TCQ modes basis will be † † H = HTCQ + ωca a + ∑ g±(ab± + h.c.), (4.27) ± with HTCQ given in Eq. 4.20. We point out that in order to obtain this model it is not necessary that both transmons are physically coupled to a resonator via a capacitance for instance. If we consider that only one transmon is coupled we would obtain the same model, but the coupling parameters would not be independent of each other. Applying the Bogoliubov transformation we find a Hamiltonian of the 72 Chapter 4. Tunable Coupling Qubit for parity measurements

2ω ˜ + + δ˜+ |2+0−i |1+1−i ω˜ + + ω˜ − + δ˜c

2ω ˜ − + δ˜− |0+2−i ω˜ + |1+0−i

ω˜ − |0iL = |0+1−i √ 2g b˜ +

|1iL = |0+0−i

FIGURE 4.6: First six energy levels and transitions of a TCQ. The states in which the qubit is encoded are shown in red. same form but with modiﬁed couplings coefﬁcients compared to the bare ones

˜ ˜ † ˜† H = HTCQ + ωca a + ∑ g˜±(ab± + h.c.), (4.28) ± with the new coupling coefﬁcients

g˜± = g+ cos λ ∓ g− sin λ. (4.29)

If the bare transmons are equal λ → −π/4, and also the bare couplings are equal √ g+ = g− = g, we get g˜+ = 2g, g˜− = 0. This situation is depicted in Fig. 4.6, where the transitions that can be caused by the resonator are shown. In this case the mode with annihilation operator b˜−, which is the mode associated with our qubit is not coupled to the resonator and consequently it cannot decay via the resonator. We say that this mode is dark with respect to the resonator. Surprisingly this does not mean that we cannot measure the state of the qubit. The reason can be qualitatively understood from Fig. 4.3. Although our qubit transition |0+0−i ↔ |0+1−i is not coupled to the resonator, the levels themselves are still coupled via the transitions involving the second (bright) mode of the TCQ. By applying a dispersive SW transformation like in Sec. 4.1, we then expect that the state |0+0−i will cause a dispersive shift of the cavity frequency χ0+0− induced by the transition |0+0−i ↔ |1+0−i, while |0+1−i will give a shift χ0+1− induced by the transition |0+1−i ↔ |1+1−i. There is no reason to believe that these shifts are equal to each other, so the states can be distinguished. However, the qubit, since not coupled to the resonator, cannot be driven via it. In order to perform gates the TCQ needs to be moved from the protected conﬁguration. This can be done dynamically by using SQUIDs in place of the single junctions and by controlling the ﬂux on the SQUIDs [123]. 4.3. TCQ for direct three-qubit parity measurements 73

0.50

0.25 ˜ 0 g-/Ec

-0.25

-0.50

(A)

13 ˜ ω-/ 12 EC

(B)

FIGURE 4.7: A) coupling strength to the resonator and B) frequency of the qubit as a function of the Josephson energies in a TCQ. We assumed the bare capacitances of the two transmons to be equal.

4.2.2 Tunable coupling

We now consider directly the situation in which we have SQUID loops. In this case the Josephson energy of the two transmons can be tuned by controlling the ﬂux in the loops. Considering a TCQ coupled to a resonator as discussed in the previous subsection, this allows to control the coupling to the resonator and the frequency of the qubit independently [122]. We show this feature in Fig. 4.7. We clearly see that we can move in the parameter space keeping the coupling coefﬁcient constant, while modifying the frequency, or vice versa.

4.3 TCQ for direct three-qubit parity measurements

We now come to the main part of this chapter in which we analyze the use of the TCQ as a qubit for the three-qubit parity measurement scheme described in Sec. 3.3. We will perform the analysis considering only one TCQ coupled to two resonators like we did for the analysis of the single transmon in Sec. 4.1. The main reason for 74 Chapter 4. Tunable Coupling Qubit for parity measurements proceeding in this way is to keep the number of parameters and terms in the Hamil- tonian small, allowing us to focus on the main effects. Additionally, if we consider three TCQ whose frequencies are far away from each other and if the coupling parameters are not too strong, we can assume that the physics does not change too much when we couple more TCQ to the resonators. We proceed similarly to Subsec. 4.2.1 and consider a TCQ that is now coupled linearly to two bosonic modes

2 2 † † H = HTCQ + ∑ ωkak ak + ∑ ∑ gk± akb± + h.c. , (4.30) k=1 k=1 ± with HTCQ again the Hamiltonian of the TCQ given in Eq. 4.20 in the bare basis. Applying the Bogoliubov transformation introduced in Sec. 4.2, we get

2 2 ˜ ˜ † ˜† H = HTCQ + ∑ ωkak ak + ∑ ∑ g˜k± akb± + h.c. , (4.31) k=1 k=1 ± where in the new basis we have effective coupling parameters that are a linear combination of the bare ones, namely

g˜k± = gk+ cos λ ∓ gk− sin λ. (4.32)

As shown in Fig. 4.8, in general the two resonators can cause the same transitions. However, the situation in which we are interested in is the one in which this is not the case as we argued in Sec. 4.1. We would like now to obtain a dispersive, effective Hamiltonian for the first two levels of the TCQ using a SW transformation. It is important to notice that in order to obtain a correct description in second order SW of the first three levels of the TCQ, it is necessary to perform the transformation considering the first six levels, and then projecting onto the subspace. We are actually interested in the first two levels, so that we could in principle omit the level |2+0−i from the beginning, but we will continue to obtain the general result for the first three levels. We thus start by projecting the Hamiltonian H˜ onto the subspace spanned by the first six levels 2. We can write

H˜ = H˜ 0 + V˜int, (4.33) with the diagonal part of the Hamiltonian

H˜ 0 = ω˜ + |1+0−i h1+0−| + ω˜ − |0+1−i h0+1−| + (2ω˜ + + δ˜+) |2+0−i h2+0−| + 2 ˜ ˜ † (2ω˜ − + δ−) |0+2−i h0+2−| + (ω˜ + + ω˜ − + δc) |1+1−i h1+1−| + ∑ ωkak ak, (4.34) k=1

2In order to keep the notation simple, we will continue to call the projected Hamiltonian H˜ . 4.3. TCQ for direct three-qubit parity measurements 75

2ω ˜ + + δ˜+ |2+0−i |1+1−i ω˜ + + ω˜ − + δ˜c

2ω ˜ − + δ˜− |0+2−i ω˜ + |1+0−i

ω˜ − |0+1−i

Res 1

Res 2 |0+0−i

FIGURE 4.8: First six energy levels of a TCQ with transitions that can be caused by the two resonators. In the general case the resonators can cause the same transitions. and the off-diagonal part that described the interaction

2 √ ˜ Vint = ∑ g˜k+ ak |1+0−i h0+0−| + |1+1−i h0+1−| + 2 |2+0−i h1+0−| + h.c. + k=1 √ g˜k− ak |0+1−i h0+0−| + |1+1−i h1+0−| + 2 |0+2−i h0+1−| + h.c. . (4.35)

In analogy with Sec. 4.1 we want to ﬁnd the generator of the second order SW transformation S1 such that

[S1, H˜ 0] = −V˜int. (4.36)

We deﬁne the detunings ∆˜ k± = ω˜ ± − ωk and consider the case in which all the interactions are dispersive, which means that they satisfy the conditions |g˜ /∆ | 1, √ k± k± | 2g˜k±/(∆˜ ± + δ˜±)| 1, and |g˜k±/(∆˜ ± + δ˜c)| 1. The operator S1 is obtained as

2 g˜k+ g˜k+ S1 = ∑ ak |1+0−i h0+0−| + |1+1−i h0+1−| ∆˜ k+ ∆˜ k+ + δ˜c k=1 √ 2g˜k+ g˜k− + |2+0−i h1+0−| + |0+1−i h0+0−| + ∆˜ + δ˜ ∆˜ k+ + √ k− g˜k− 2g˜k− |1+1−i h1+0−| + |0+2−i h0+1−| −h.c.. (4.37) ∆˜ k− + δ˜c ∆˜ k− + δ˜−

The effective Hamiltonian for the ﬁrst six levels is given H˜ eff = H˜ 0 + [S1, V˜int]/2. The rather long full result can be found in Appendix A of Ref. [76]. By ﬁnally projecting 76 Chapter 4. Tunable Coupling Qubit for parity measurements

onto the qubit subspace spanned by |1iL = |0+0−i and |0iL = |0+1−i we obtain

H˜ d = Ω˜ − |0+1−i h0+1−| + 2 + | i h | − | i h | † + ∑ ωk χk,0+1− 0+1− 0+1− χk,0+0− 0+0− 0−0+ ak ak k=1 † χ12,0+1− |0+1−i h0+1−| − χ12,0+0− |0+0−i h0+0−| a1a2 + h.c. , (4.38)

where χk,0+1− and χk,0+0− are the dispersive shifts of the resonator k caused by the states |0+1−i and |0+0−i respectively. Moreover, χ12,0+1− represents the coupling between the two resonators if the qubit is in the state |0+1−i, while χ12,0+0− if the state is |0+0−i. These parameters are explicitly given by the following formulas √ 2 2 2 g˜k− ( 2g˜k−) g˜k+ χk,0+1− = − − , (4.39a) ∆˜ k− ∆˜ k− + δ˜− ∆˜ k+ + δ˜c

2 2 g˜k+ g˜k− χk,0+0− = + , (4.39b) ∆˜ k+ ∆˜ k−

√ √ g˜1−g˜2− 1 1 ( 2g˜1−)( 2g˜2−) 1 1 χ12,0+1− = + − + − 2 ∆˜ 1− ∆˜ 2− 2 ∆˜ 1− + δ˜− ∆˜ 2− + δ˜− g˜ +g˜ + 1 1 1 2 + (4.39c) 2 ∆˜ 1+ + δ˜c ∆˜ 2+ + δ˜c g˜1+g˜2+ 1 1 g˜1−g˜2− 1 1 χ12,0+0− = + + + . (4.39d) 2 ∆˜ 1+ ∆˜ 2+ 2 ∆˜ 1− ∆˜ 2− The renormalized energy of the ﬁrst excited state is obtained as

2 ˜ 2 Ω− = ω˜ − + ∑ g˜k−/∆k−. (4.40) k=1

z z Introducing now the Pauli σ operator as |0+0−i h0+0−| = (1 − σ )/2, |0+1−i h0+1−| = (1 + σz)/2, we rewrite the Hamiltonian Eq. 4.38 as

˜ 2 Ω− z z z † Hd = σ + ∑ (ω¯ k + χkσ ) + (χ¯12 + χ12σ )(a1a2 + h.c.), (4.41) 2 k=1 which is of the same form as Eq. 4.11, but with modiﬁed parameters. Speciﬁcally

χk − χk ω¯ = ω + ,0+1− ,0+0− , (4.42a) k k 2

χk + χk χ = ,0+1− ,0+0− , (4.42b) k 2 χ − χ χ¯ = 12,0+1− 12,0+0− , (4.42c) 12 2 4.3. TCQ for direct three-qubit parity measurements 77

χ + χ χ = 12,0+1− 12,0+0− , (4.42d) 12 2 k = 1, 2.

4.3.1 Achieving zero χ12

From Eqs. 4.39 and 4.42 it is evident that the TCQ gives more freedom in the parameter choice compared to a simple transmon. In particular, consider the case in which

g˜1+ 6= 0 , g˜2− 6= 0, (4.43a)

g˜1− = g˜2+ = 0. (4.43b)

In this case, we get that both the quantum switch coefﬁcient χ12 and also the induced

fixed coupling between the resonator modes χ¯12 are zero. This situation is shown in Fig. 4.9. In practice, in this case resonator one can cause only transitions associated with the "+" mode of the TCQ, while resonator two only transitions related to the "−" mode. Notice the analogy with Fig. 4.2 discussed for the transmon qubit. We can say that in this configuration the qubit is dark for resonator 1, but not for resonator 2. However, the qubit can be measured by both resonators, since both resonators get a dispersive shift that depends on the state of the qubit. This is basically due to the same property that allows the TCQ to be measured even when it is Purcell protected in the single resonator case, as discussed in Subsec. 4.2.1. We point out however that in the configuration in which the quantum switch term is zero the qubit is not Purcell protected, since it can decay through resonator 2. Let us now try to understand what we need in order to realize the condition discussed above. Considering the bare transmons composing the TCQ to be equal (λ = −π/4), we obtain from Eq. 4.32 the following conditions on the bare coupling coefficients that satisfies Eqs. 4.43

g1+ = g1− = g1, (4.44a)

g2+ = −g2− = g2, (4.44b) √ √ which give additionally g˜1+ = 2g1, g˜2− = 2g2. From Eqs. 4.44 we see that in order to have zero quantum switch, we need to be able to engineer a sign change of the coupling parameters to resonator 2. This ability is non-trivial to achieve pas- sively. We now focus on how this can be achieved practically using transmission line resonators. In fact, it is not possible to obtain this condition considering the resonators to be lumped elements. This would be possible if we had available another lumped element, ı.e., a transformer with negative turns ratio (see Subsec. 2.4.1), but unfortunately it is quite inconventient to have a transformer in a superconducting integrated circuit. The idea is then to exploit the distributed character of a transmission line resonator. The point is that the sign of the coupling coefﬁcient depends on the position of the coupling capacitance. In particular, as shown in AppendixA, 78 Chapter 4. Tunable Coupling Qubit for parity measurements

2ω ˜ + + δ˜+ |2+0−i |1+1−i ω˜ + + ω˜ − + δ˜c

2ω ˜ − + δ˜− |0+2−i ω˜ + |1+0−i

ω˜ − |0+1−i

Res 1

Res 2 |0+0−i

FIGURE 4.9: Transitions that give zero quantum switch and fixed coupling between the two resonators. if we consider that the relevant mode is the n-th mode of the resonator 3, we can consider the flux field Φ(x, t) to be given by

√ πn Φ(x, t) ≈ 2 cos x Φn(t), (4.45) L0 with L0 the length of the transmission line and Φn(t) a dynamical variable, which in the decoupled√ case satisﬁes the dynamics of a harmonic oscillator with frequency ωn = πn/( `cL0). If we consider a transmon capacitively coupled to a certain position xT of a transmission line resonator via a capacitance Cc we then expect a term in the Lagrangian

2 √ 2 Cc ∂Φ(xT, t) Cc πn Φ˙ T − = Φ˙ T − 2 cos xT Φ˙ n , (4.46) 2 ∂t 2 L0 with ΦT the node variable representing the transmon mode. It is clear that then the sign of the coupling coefficient depends on the position of the transmon. We point out that this is true at the Lagrangian as well as at the Hamiltonian level. Based on this simple idea, we can understand that in order to have zero χ12 we need three coupling capacitances to couple positively and one negatively, or vice versa. This means that the bare transmons composing the TCQ need to be coupled at different positions. In this configuration the TCQ is quite similar to what has been referred in the literature as a giant atom [127, 128], which couples at different postions of a transmission line. Based on this idea we show in Fig. 4.10 a possible implementation of this scheme considering the TCQ to be coupled to transmission line resonators. In particular, the system is engineered to have the necessary sign flip of the interaction coefficients. We show the figure considering the mode n = 3 for resonator 2, in which we have the sign flip, while mode n = 2 for resonator 1 in which no sign flip is needed. The reason why it might be preferable to use the n = 3 mode for the resonator in which the sign flip needs to be engineered is that it presents more sign

3We take open circuit boundary conditions here. 4.3. TCQ for direct three-qubit parity measurements 79

++ ---- + + + + -- n = 3

n = 2

++ ---- ++

FIGURE 4.10: Possible three-qubit parity measurement setup achieving zero quantum switch term.

Cc CI Cc Φ1 Φ+ Φ− Φ2

C1 L1 CJ+ EJ+ CJ− EJ− C2 L2

FIGURE 4.11: TCQ with the two bare transmons coupled capacitively only to one resonator. The capacitances of the bare transmons are taken to be equal for simplicity changes. This allows us to avoid the crossing of wires as shown in Fig. 4.10.

4.3.2 Case with χ12 6= 0

In the previous subsection we have described the case in which χ12 = 0 and how this configuration can be engineered. We know however, that although this might be considered the ideal case, in principle we do not need to satisfy this condition ex- 2 actly but only to ensure that χ1χ2 ≥ χ12. We now show how this condition can also be satisfied considering the case in which one of the two transmons is coupled capacitively to resonator 1, while the other one to resonator 2. This configuration is shown in Fig. 4.11. It is important to point out that this does not correspond exactly to the case in which g2+ = 0, g1− = 0. This is actually never the case, since proceeding with the standard circuit quantization of the circuit in Fig. 4.11, the physical locality of the capacitances is not preserved in the inversion of the capacitance matrix, and for this reason we get also a coupling between resonator 1 and bare transmon "−", as well as between resonator 2 and transmon "+". These interactions can be neglected in the case in which CI CJ, CI C1,2. However, since in our case we consider CI to be several fractions of CJ this contribution has to be taken into account. In Fig. 4.12 we 2 show how we can indeed achieve the regime in which χ1χ2 ≥ χ12. The parameters 80 Chapter 4. Tunable Coupling Qubit for parity measurements

2 0.000005 1 2/EC 2 2 12/EC 0.000004

0.000003

0.000002

0.000001

0.000000 0.1 0.2 0.3 0.4 0.5 CI/CJ

2 FIGURE 4.12: χ1χ2 and χ12 as a function of the coupling capacitance CI. The data are shown taking EC as a unit (¯h = 1). used to generate the plot are speciﬁed in Table 4.1. Just for illustration purposes we assumed the resonators to be equal, although clearly this is not the condition that allows a three-qubit parity measurement to be realized. However, this deﬁnitely does 2 not change the possibility of achieving χ1χ2 ≥ χ12 since the resonators should be only slightly detuned.

EC,bare/h 300 MHz

EJ 60EC,bare

Cc 0.1CJ

Cr 10.36CJ

Lr 0.07LJ

TABLE 4.1: Parameters for Fig. 4.12. We assumed equal transmons CJ+ = CJ− = CJ, EJ+ = EJ− = EJ and also equal resonators 2 2 2 C1 = C2 = Cr, L1 = L2 = Lr. LJ = Φ0/(4π EJ), EC,bare = e /(2CJ). These are realistic parameters considering the resonators to be transmission line resonators with characteristic impedance Z0 = 50 Ω. Ad- ditionally, all necessary approximations are well satisﬁed.

This conﬁguration is deﬁnitely easier to implement compared to the case in which

χ12 = 0. However, it has the disadvantage that, when considering three TCQ, we have to consider that also the parameter χ12 has to be matched between the three qubits. A possible conﬁguration considering three TCQ coupled to two transmission line resonators is shown in Fig. 4.13. Like for the case with χ12 = 0, we point out that we lose the possibility to have a Purcell protected qubit in this case.

4.3.3 Black-box approach

In the previous sections we have shown a possible way to engineer the required interactions in our three-qubit parity measurement scheme based on the coupling to 4.3. TCQ for direct three-qubit parity measurements 81

++ ++ n = 2 ----

FIGURE 4.13: Possible conﬁguration that allows a three-qubit parity 2 measurement with χ1χ2 ≥ χ12, with χ12 6= 0 using the modes n = 2 of two transmission line resonators. transmission line resonators. Of course the use of transmission lines is only one possibility of the most general problem that is that of designing a microwave network seen by the qubits, that gives the desired properties. We sketch here the approach that has been implemented in Ref. [129]. The idea is to obtain the impedance matrix that implements correctly the Hamiltonian that was previously described. To connect the impedance matrix to the Hamiltonian formalism we use Foster’s representation of a multiport, lossless and reciprocal network [39, 40, 130, 36], as shown in Fig. 4.14b. In particular, we consider directly the case in which we have two relevant modes. Impedance matrices with more modes, like for the case of transmission lines, should approximate the ideal two-mode impedance close to the relevant frequencies. Notice that the network depicted in Fig. 4.14b is a six-port network, while the total network is seven-port, since we should add the drive port. However, it is easier to reason in terms of the six-ports and then extend it to the seven-port. We will only treat the six-port here. In addition, we have not included the coupling capacitances in the network. The sign change in the Jaynes-Cummings coupling parameters in this representation is obtained simply by a sign change of the turns ratios of the transformers (see Subsec. 2.4.1). A possible scheme is to simply use turns ratios +1 and −1. Only one resonator needs the sign change. Once we know the turns ratios we can directly obtain the impedance matrix of the network as

2 1 ω2 Z[ω] = B , (4.47) ∑ 2 2 k k=1 iωCk ω − ωk 82 Chapter 4. Tunable Coupling Qubit for parity measurements

Z[ω]?

(A) Three TCQ coupled to a linear black-box represented by an impedance matrix Z[ω].

(B) General Foster representation of the network with two modes (parameters are not shown).

FIGURE 4.14: Black-box parity network. 4.3. TCQ for direct three-qubit parity measurements 83

with Ck, the equivalent capacitances of the resonators and ωk their frequencies. The matrices B1,2 are the turns ratios matrices of resonator 1 and 2   1 1 1 1 1 1   1 1 1 1 1 1     1 1 1 1 1 1 B1 =   , (4.48a) 1 1 1 1 1 1   1 1 1 1 1 1   1 1 1 1 1 1

  1 −1 1 −1 1 −1   −1 1 −1 1 −1 1    − − −   1 1 1 1 1 1 B2 =   . (4.48b) −1 1 −1 1 −1 1    1 −1 1 −1 1 −1   −1 1 −1 1 −1 1 More details can be found in Ref. [129].

Chapter 5

Hamiltonian quantum computing

This chapter contains material by the author from Ref. [131] that is in preparation for publi- cation.

5.1 Different models of quantum compuation

When the quantum information scientist thinks about how a working quantum computer could ever be realized, in most cases, he is actually unconsciously thinking about a particular model of quantum computation, that is the circuit model of quantum computation [4]. In the circuit model of quantum computation we have N qubits with an associated Hilbert space H = C2 ⊗ C2 ⊗ · · · ⊗ C2, with C2 the Hilbert space of a single qubit 1. In each qubit Hilbert space we identify a computational basis {|0i , |1i}, which natually induces a computational basis in the full Hilbert space

H. The qubits are initialized in a certain state |ψini, which without loss of generality can be taken to be |ψini = |00 . . . 0i. The quantum computation is then described by a collection of unitary operators {U1, U2,..., UM} acting on the state |ψini to give a collection of intermediate states {|ψ0i , |ψ1i ,..., |ψMi} with |ψki = UkUk−1 ... U1 |ψini. In general, the unitaries we choose act non-trivially on a limited number of qubits, usually no more than two or three, depending on the gate set we are using. The result of the computation is extracted by measuring the final state |ψfini = |ψMi in the computational basis, a procedure that can be done by measuring each qubit sep- arately. The complexity of an algorithm in the circuit model is usually quantified in terms of the number of gates necessary to implement the algorithm. In this abstract setting the circuit model does not assume anything about how these unitaries are implemented, but it is quite natural to think that they are implemented in time one after the other. We can say that this is the standard approach to quantum computing. Quantum algorithms and quantum error correction are mostly conceived with these ideas in mind. The unitaries are obtained by exponentiating a certain Hamiltonian, where in case of time-dependent Hamiltonian, we mean time-ordered exponential of course. In the Hamiltonian we usually have parameters that we can turn on and

1We mean here the restriction of these Hilbert spaces to normalized states of course. 86 Chapter 5. Hamiltonian quantum computing off in time so that the desired unitary is implemented. For the specific case of superconducting qubits these parameters are basically fluxes in the loops and microwave drives. While for the quantum physicist it is quite natural to think in terms of the circuit model of quantum computation, as it also happens in classical computation, there are several equivalent models of quantum computation. We say that two models for quantum computation, say a and b, are equivalent if we can simulate whatever algorithm designed for model a, with model b, with at most a polynomial overhead (and vice versa). In the remaining part of this section, we briefly describe the basic ideas of some of these models for quantum computation. Maybe the most popular alternative model for quantum computation is the model of adiabatic quantum computation (AQC) [132, 133]. Adiabatic quantum computation is based on a fundamental theorem of quantum mechanics that is the adiabatic theorem. Roughly speaking the adiabatic theorem says that if a system is prepared in the ground state of a Hamiltonian Hin at time t = 0 and the Hamiltonian is slowly transformed in time into another Hamiltonian Hfin at time tfin, the system ends up in the ground state of Hfin. What one means by slowly here is that the evolution should be slow compared to the inverse of the gap of the Hamiltonian, ı.e., the energy difference between first excited and ground states. The idea of AQC is then to prepare the ground state of an easy Hamiltonian Hin of a system of N qubits and evolve it via a certain adiabatic routine to the ground state of Hfin. The ground state of Hfin embeds the result of the computation that is then read out by measuring in the computational basis. In general, the ground state of Hin is trivial, meaning that it can be one of the computational basis states, while we expect the ground state of Hfin to present some sort of quantum entanglement. It is worth mentioning a subclass of AQC that is quantum annealing (QA) [134]. A quantum annealer is a non-universal adiabatic quantum computer specifically designed to solve combinatorial optimiza- tion problems. While the role played by QA in quantum complexity theory, or from a more practical point of view its usefulness, is still unclear, it is the working principle of the D-Wave quantum computer [135], which is to date the only commercially available computer which deserves the adjective quantum to be used. Another model of quantum computation is instead based on the existence in some two-dimensional systems of quasiparticles that are neither bosons nor fermions, but are known as non-Abelian anyons. This setting defines the field of topological quantum computation [136, 137]. The standard physics that hosts these exotic excitations is that of the fractional quantum hall effect (FQHE). Despite the practical difficulties in manipulating these kinds of systems, topological quantum computation holds the promise to have an in-built resiliance to environmental noise, due to its topological nature. In the so-called one-way quantum computation, also known as measurement based quantum computation, the quantum algorithm is implemented in a rather special way [138, 139]. This paradigm of quantum computation relies on the initial 5.2. Feynman quantum computer 87 preparation of highly entangled states, called cluster states. This step is algorithm independent. The algorithm is then implemented by a specific sequence of single- qubit measurements on this initial state. In a certain sense, we can say that a cluster state has inside it all the computational power of quantum computation. In the remaining part of this section we will consider models of quantum computation based on time-independent Hamiltonians. The general paradigm of these models is the following. The system is initialized in an easily prepared state. The system evolves under the action of a time-independent Hamiltonian for a certain time. The system is measured in the computational basis and the result of the computation is extracted. Among these models we mention discrete-time and continuous-time quantum walk [140, 141, 142, 143]. It is not by chance that the reader might find a similarity with AQC. In fact, some of these models can be adapted to obtain an adiabatic version of them.

5.2 Feynman quantum computer

It is rather surprising that already in the eighties there were alternative models of quantum computation, in particular a model constructed by Feynman in a seminal paper [18]. The Feynman model is, as we will see, closely connected to the circuit model, but there are two fundamental differences:

1. the quantum computation is achieved explicitly with a time-independent Hamit- lonian;

2. the presence of an auxiliary system that plays the role of the clock.

We will now introduce what we will call the Feynman Hamiltonian, starting from a certain algorithm described in the circuit model language. Suppose we have a system of N qubits, which we call the data system. Its 2N-dimensional Hilbert space is denoted by Hdata. We want to implement a generic quantum algorithm that in the circuit model is represented by a sequence of M unitary gates {U1,..., UM}, starting from an initial state |ϕini ∈ Hdata. In order to achieve this with a time- independent Hamiltonian, we introduce an additional system, that we will call the M+1 clock, deﬁned on a M + 1-dimensional Hilbert space Hclock = C . We consider a particular clock basis {|ki , k = 0, 1, . . . , M}. The total Hilbert space we consider is

H = Hdata ⊗ Hclock. The Hamiltonian that Feynman considered was

M HF = −J ∑ Uk |ki hk − 1| + h.c., (5.1) k=1 where the operator |ki hk − 1| and its hermitian conjugate act on the clock, while Uk † and Uk on the data. The parameter J, takingh ¯ = 1, has the units of frequencies and in what follows we will always take J > 0 2. It is quite interesting to understand in

2Taking J < 0 does not change the problem anyway. 88 Chapter 5. Hamiltonian quantum computing which sense Eq. 5.1 implements a quantum computation. To this end it is convenient to introduce the intermediate computational states

|ψki = UkUk−1 ... U1 |ϕini |ki , (5.2) k = 0, 1, . . . , M. Notice that |ψ0i = |ϕini |0i = |ψini, while we are interested in the information encoded in the ﬁnal computational state |ψﬁni = |ψMi. The intermediate computational states are orthogonal and they span a M + 1-dimensional subspace of the total Hilbert space H. We call this subspace the computational subspace Hcomp ⊂ H. We will now show that this subspace is invariant under the action of the Feynman Hamiltonian. This means that if the system starts in a generic state |ψi ∈ Hcomp, then the evolved state |ψ(t)i = exp[−iHFt] |ψi ∈ Hcomp, ∀t. We can prove this by showing that the action of HF on a state |ψki given by Eq. 5.2 always produces a linear combination of intermediate computational states. In fact for k 6= 0, M

HF |ψki = −J(|ψk+1i + |ψk−1i), (5.3) while we get at the boundaries

HF |ψ0i = −J |ψ1i , HF |ψMi = −J |ψM−1i . (5.4)

This implies that if at t = 0 the state is |ψ0i we can always write

M Pcomp = ∑ |ψki hψk| (5.6) k=0 and show that it commutes with HF, ı.e., [HF, Pcomp] = 0, which is also a consequence of the previous calculations. From Eq. 5.5 we see that in Feynman’s construction the system evolves in time in a superposition of all the intermediate computational states. Without entering into details about the efﬁciency of the scheme, in order to extract the result of the quantum computation the idea is to measure the clock at a certain time. If we measure the clock in the ﬁnal state |Mi we thus proceed with the measurement of the data. Notice that in principle there is a small probability to measure the clock in state |Mi at any time t. There is also an additional consequence of the previous discussion and in particular of Eq. 5.3. If we start in a state |ψi ∈

Hcomp, we can restrict the Hamiltonian to this subspace to obtain

M HF|comp = Pcomp HFPcomp = −J ∑ |ψki hψk−1| + h.c.. (5.7) k=1 5.2. Feynman quantum computer 89

This Hamiltonian can be interpreted as the Hamiltonian of a particle hopping on a discrete line with M sites, with hopping coefficients J. The model we obtained is what a theorist in the field of quantum information would call a continuous- time quantum walk on one line, while a condensed matter physicist would call a one-dimensional tight-binding model. Being the simplest quantum walk or tight- binding model that one can think of, it turns out that it has an analytic solution, which allows us to analyze the dynamics of the computation. We postpone this analysis to Sec. 5.5, when we will introduce a model that is closely related to the Feyn- man Hamiltonian. There is an additional important feature that we have obtained, that’s to say that the dynamics does not depend at all on the particular unitary gates we are implementing. All these features of the Feynman Hamiltonian will be shared by the other models we will deal with in this chapter. We finally point out that in principle there is no reason to restrict ourselves to the case in which the hopping constant J is indeed a constant. We could consider different Jk for every transition and the previous reduction of the dynamics would be the same. This however can allow us to design a more efficient dynamics of the system. These ideas are the same as those employed in the context of perfect quantum state transfer [144, 145], and originally employed by Peres for the Feynman Hamiltonian in Ref. [146]. For further modern discussions about the Feynman Hamiltonian we point the reader to Refs. [147, 148].

5.2.1 Clock representation

A fundamental element in the Feynman construction is the clock. In the previous section we have deﬁned the clock as a system living in a M + 1-dimensional Hilbert space Hclock. Considering that the unitaries that we need to apply to achieve a universal gate set can be taken to be at most two-qubit unitaries, we are tempted to say that the Feynman Hamiltonian requires at most three-body terms. A typical three- body term would apply a two-qubit unitary on two qubits, and also shift the clock state. However, this is not a fair evaluation from the point of view of complexity theory. In fact we are considering on the same footing a qubit and a M + 1-dimensional system, while it would be better to analyze a system with qubits only. We can quite easily conceive a clock made out of qubits. Let us consider a system of M + 1 qubits. We can construct a clock analogous to the previous one by restricting this system to the single excitation subspace. A clock state |ki, with k = 0, 1, . . . , M would be then represented by |ki → |0 . . . 0 1 0 . . . 0i (5.8) | {z } | {z } k M−k with the one placed at the k + 1-th position. This kind of clock is known as pulse clock. With this representation of the clock the Feynman Hamiltonian assumes the form M HF,P = −J ∑ Uk |01i h10|k−1,k + h.c., (5.9) k=1 90 Chapter 5. Hamiltonian quantum computing and we see from the previous discussion that it can involve at most four-body terms, since the clock terms are all two-body. We actually say that the Feynman Hamilto- nian with pulse clock is 4-local and in general we deﬁne a k-local operator as [149]:

Deﬁnition 5.2.1. An operator O acting on N qubits is k-local if it can be written as O = r ∑j=1 Oj where each term Oj is hermitian and acts non-trivially on at most k qubits.

If our operator is a Hamiltonian we talk about a k-local Hamiltonian. It is important to notice that the locality of a k-local Hamiltonian has nothing to do with spatial locality. It is rather a locality in the sense of weight of the interactions involved. The pulse clock is not the only possible clock construction that can be obtained from a system of M + 1 qubits. Another popular construction is the domain-wall clock in which we map |ki → |11 . . . 1 00 . . . 0i . (5.10) | {z } | {z } k+1 M−k With the domain-wall clock the Feynman Hamiltonian becomes

M HF,DW = −J ∑ Uk |110i h100|k,k+1,k+2 + h.c., (5.11) k=1 and in this case since the clock part of the Hamiltonian involves three-body terms, the Feynman Hamiltonian involves at most ﬁve-body terms and it is thus 5-local.

5.2.2 Kitaev construction and the local Hamiltonian problem

The Feynman approach to quantum computation has received little attention as a way of realizing a quantum computer in practice. The main reason for this is that there is an immediate bottleneck that is the fact that the interactions involved need to be either 4-local or 5-local, depending on the clock implementation. Interactions with high weight are usually difficult to implement in any physical implementation of a quantum computer, but they can be obtained perturbatively starting from two-body interactions. There is however an additional practical problem, with the Feynman construction. It does not seem possible to construct Feynman computer keeping the interactions needed spatially local. In fact, although the quantum algorithm can be designed to involve only nearest-neighbour gates, the interaction with the clock qubits will eventually require coupling between data and clock qubits that are far away from each other. At a theoretical level instead the Feynman quantum computer has lead to important results. It is quite intuitive to understand that the Feynman construction is a strong candidate to realize a connection between the circuit model and adiabatic quantum computation. In fact, it is straightforwardly related to the circuit model and it shares with AQC the feature that the computational problem is embedded in a Hamiltonian. This connection has been fully exploited in the proof of the equivalence between the circuit model and AQC given in Ref. [150]. Other approaches to 5.2. Feynman quantum computer 91 the proof rely also on the Feynman Hamiltonian or on closely related constructions [151, 152]. The first to realize the theoretical importance of Feynman construction was Ki- taev in Ref. [153], where he used Feynman’s idea to prove the QMA-completeness of the k-local Hamiltonian problem. QMA (Quantum Merlin Arthur) is the quantum version of the classical computational class MA, which is in turn the probabilistic version of the more famous computational class NP. The k-local Hamiltonian problem is a promise problem in which we have to decide whether the minimum eigenvalue of a k-local Hamiltonian is below a certain value a or above a certain b, with b > a, given the promise that indeed one of the two conditions is satisfied. For more details about the QMA-completeness of the k-local Hamiltonian problem we direct the reader to the original derivation in Ref. [153], and also to related literature [149, 152]. What we want to show now is just how we can adapt Feynman construction to encode the result of a computation in the ground state of a Hamiltonian. This is the trick used by Kitaev to formulate any verifier circuit in terms of a k-local Hamil- tonian problem and also the basic idea of the proof of the equivalence between AQC and circuit based quantum computation. We start by modifying the Feynman Hamiltonian Eq. 5.1 following Kitaev [153]

M M ˜ † HK = J ∑ {|k − 1i hk − 1| + |ki |ki − |ki hk − 1| Uk − |k − 1i hk| Uk } = ∑ Hk, k=1 k=1 (5.12) where again we take J > 0. Before analyzing the previous Hamiltonian, let us suppose that in our quantum circuit we start the computation with the data in one of the 2N states that form the computational basis. We compactly denote them by |xi, x ∈ {0, 1, . . . , 2N − 1}, where we mean the binary representation of x. We deﬁne the history computational state with initial data state |xi as

1 M x √ |ψhisti = ∑ |ki UkUk−1 ... U1 |xi . (5.13) M + 1 k=0

We have consequently 2N orthogonal history states. The history states are basically an equal superposition of all the intemediate computational states. In each of them the probability of measuring the ﬁnal computational state is equal to 1/(M + 1). We now show that the history states are all ground states of the Hamiltonian Eq. 5.12 N and thus H˜ K has a 2 degenerate ground space. In order to show this we notice that

H˜ K is a positive semi-deﬁnite operator. In fact each Hk is a positive semi-deﬁnite † 2 operator since Hk = Hk and Hk = 2JHk. It is then immediate to show that for x each history state and each Hk, Hk |ψhisti = 0. This shows that the history states are eigenstates of H˜ K with eigenvalue 0, which implies that they form the groundspace since H˜ K ≥ 0. This also implicitly shows that H˜ K is a frustration free Hamiltonian, meaning that the groundspace is groundspace of each of its elements. 92 Chapter 5. Hamiltonian quantum computing

While we have seen that the history states deﬁned in Eq. 5.13 form the degenerate groundspace of H˜ K, we actually would like a particular history state to be the 0 unique ground state, for instance |ψhisti. In order to do this we just need to modify the Hamiltonian H˜ K and give penalties to states with data qubits not initialized in |00 . . . 0i. We then consider a new Hamiltonian

M ˜ HK = HK + ∑ |1i h1|k |0i h0|clock . (5.14) k=1

0 The last term does exactly the job we have just described and HK has |ψhisti as unique ground state. This construction allows to connect a generic quantum circuit with the groundstate of a Hamiltonian. It should be noted that the ground state is not the ﬁnal computational state, but the history state, and so the result of the computation is always obtained probabilistically.

5.3 Hamiltonian quantum computing on a lattice

In this section we start to enter the main part of this chapter and consider the model for quantum computation with a time-independent Hamiltonian introduced in Ref. [19] (see also related works). After reviewing the basic idea of the scheme we show explicitly how to construct a Toffoli gate in this scheme involving nearest-neighbour encoded qubits. In the following section we will introduce a related model which has a different dynamics, which turns out to be the same as that of the Feynman Hamiltonian. In summary the models we are going to describe are closely related to the Feynman construction, with some important differences:

1. the basic Hamiltonian is only 2-local;

2. the effective Hamiltonian is 4-local and obtained perturbatively (perturbative gadget);

3. there is no explicit clock system;

4. the interactions are sufﬁciently local in space;

As we will see there is a price to pay for these advantages, which we can summarize as:

1. an increase in the number of qubits needed (polynomial);

2. errors introduced by the use of perturbation theory.

We start by introducing how gates are executed in these schemes with a simple example. 5.3. Hamiltonian quantum computing on a lattice 93

−JVhop,X

|Ψini

|ϕiniL |vaciR

(A) Initial state.

−JVhop,X

|Ψouti

|vaciL |ϕoutiR

(B) Final state.

FIGURE 5.1: Example of a single-qubit gate.

5.3.1 Quantum gate with a hopping particle

The main idea for implementing gates with a time-independent Hamiltonian without an explicit clock can be found in a paper by Margolus [154]. This idea was later analyzed by Mizel et.al. [155] (see also related works [156, 157]). In this paper the authors suggest to realize a qubit undergoing M single-qubit gates U1, U2,... UM in time, using a 2(M + 1)-dimensional quantum system prepared in an analogue of the history state described in Subsec. 5.2.2. Considering that this 2(M + 1)-dimensional quantum state is realized by M + 1 two-level systems, we can interpret the evolution of the computation as an evolution in space rather than in time. The model described in Ref. [155] encodes the quantum computation in the ground state of a Hamiltonian, while we will consider its dynamical version which is related to it like the Feynman Hamiltonian is related to the Kitaev Hamiltonian. Let us consider the simplest case possible in which we implement only a single-qubit gate and consider Fig. 5.1. We have two sites, left L and right R, and hopping between them. In each site we have effectively a three-level system (qutrit). For instance the left site can be empty, that’s to say be in the a state that we call the vacuum |vaciL, or host a particle with state |1iL or state |0iL. In quantum mechanics linear combinations of them are also possible and the same is valid for the right side. The previous picture can also be interpreted as a particle hopping between the two sites with an internal spin-like degree of freedom. It is in general easier to think in this last picture, but it is important to keep in mind both interpretations. Considering the case in which we want to apply a X gate, the hopping between the two sites is described by a Hamiltonian

† † HX = −JVhop,X = −J{a0[R]a1[L] + a1[R]a0[L] + h.c.}, (5.15) 94 Chapter 5. Hamiltonian quantum computing

where as[S], s = 0, 1, S = L, R annihilates a particle at site S with internal state s,

ı.e., as[S] = |siS = |vaciS. Let us suppose that the system at t = 0 is in the state

|Ψ(t)i = cin(t) |Ψini + cout(t) |Ψouti . (5.17)

Also in this case the dynamics is not inﬂuenced by the particular gate that is implemented, and it is equivalent to a quantum walk on a line with two sites. If we now consider a chain with M sites like in Fig. 5.1a, we can apply M single-qubit gates and we would obtain that the dynamics is equivalent to a quantum walk on a line with M + 1 sites. We point out that since the system is composed only by a single hopping particle, or in the qutrit picture there is only one excitation in the system, we will always have our quantum state in a linear superposition of 2(M + 1) states namely

{|0i0 |vaci1 |vaci2 ... |vaciM , |1i0 |vaci1 |vaci2 ... |vaciM ,

|vaci0 |0i1 |vaci2 ... |vaciM ,..., |vaci0 |vaci1 ... |vaciM−1 |1iM}. (5.18)

This 2(M + 1)-dimensional subspace in which our system lives can be represeted by a qubit and a M + 1-dimensional clock system in a Hilbert space H = HQ ⊗ Hclock via a unitary map W deﬁned as

W |vaci0 |vaci1 ... |0ik ... |vaciN = |0i |ki , (5.19a)

W |vaci0 |vaci1 ... |1ik ... |vaciN = |1i |ki , (5.19b) with k = 0, 1, . . . , M. We have just shown that the model of a hopping particle with two internal degrees of freedom maps unitarily to the Feynman problem in which the data qubit is a single qubit. We remark however that in the system there is no clock. The role of the clock is played by the position of the particle (excitation). If we consider many particles hopping on different lines we would have basically an equivalent clock for every particle. In this picture, if we have only single-qubit gates we can just run them in parallel, which is of course the case also for the circuit model. However, in order to run a useful quantum computation involving multi-qubit gates we need a way to coordinate these local clocks ensuring that the logic is implemented correctly. As an example, if we consider a CNOT gate, we do not want the target to 5.3. Hamiltonian quantum computing on a lattice 95

(A) Connected string.

(B) Disconnected string.

FIGURE 5.2: Examples of connected and disconnected strings of particles. The red dots denote the position of the particles. pass the CNOT region, while the control is still behind it 3. In the remaining part of this chapter we will understand how this coordination can be achieved.

5.3.2 Lattice of hopping particles

In this subsection we describe a model introduced in Ref. [19], which is closely connected to related works of Refs. [158, 159, 160]. We will describe the system using the picture of particles hopping with an internal spin degree of freedom. We start by considering the rotated N × N lattice depicted in Fig. 5.2. We denote a site or vertex by (i, j), with i, j ∈ {1, 2, . . . , N}. We will always consider the case in which there is always only one particle per track (horizontal lines) 4. This will be ensured by initial- izing the system in a state with this property and by constructing a Hamiltonian that preserves it. We will have then Ntrack = 2N − 1 tracks, ı.e., particles. A particle on a certain track can hop on different sites on the line with the application of a gate to its internal degree of freedom exactly in the same way as described in Subsec. 5.3.1. If particles on different tracks do not interact with each other what we have is just a series of independent quntum walks on a line. Interactions between particles on adjacent tracks achieve the necessary coordination to ensure that a quantum circuit

3Here we are reasoning a bit classically, but we should always keep the quantum interpretation in mind. 4A track is identified by a constant i − j. 96 Chapter 5. Hamiltonian quantum computing involving also multi-qubit gates is correctly implemented. However, at the beginning we will focus for simplicity on the case in which only single-qubit gates are implemented, and postpone the analysis of multi-qubit gates to Sec. 5.4. As we have already mentioned in this scheme the quantum information is encoded in the internal degrees of freedom of the particles. Particles can hop on different sites on a certain track and a single-qubit gate is applied on the internal degree of freedom. We can then associate a single-qubit gate with each plaquette in Fig. 5.2. We will design the dynamics of the system in such a way that the particles’ position always form a connected string like in Fig. 5.2a. This property will not be ensured exactly, but perturbatively. We will refer to these configurations as connected strings, while we will refer to configurations like the one depicted in Fig. 5.2b as disconnected strings. If we initialize the string in a state in which all particles are on the left and in a certain internal state, a certain connected string identifies univocally the gates that have been executed. Before introducing the Hamiltonian, let us introduce some mathematical defini- tions. We define the particle number operator at site (i, j) as

n[i, j] = ∑ ns[i, j], (5.20) s=0,1 where ns[i, j] is the number operator for particle at site (i, j) with internal spin state s = 0, 1. The operators ns[i, j] can be written in terms of creation and annihilation operators for a particle at site (i, j) and internal state s, ı.e.,

† ns[i, j] = as [i, j]as[i, j] (5.21)

. We additionally deﬁne the set of all vertices V, the set of edges E and the set of plaquettes PL. A vertex (i, j) will occasionally be denoted with a compact symbol µ or ν. We will identify an edge e by the sites it connects and write e = (µ, ν). The Hamiltonian is given by the sum of two fundamental terms

H = Hvalid + Vhop. (5.22)

In the following two subsections we analyze these two terms in details.

5.3.3 Valid Hamiltonian

The Hamiltonian Hvalid is the part of the Hamiltonian whose role is to ensure that the strings always remain connected, ı.e., conﬁguration that will be called valid. This means that the system evolves approximately only in a superposition of connected strings, with internal state arbitrary and depending on the particular quantum circuit we want to implement. It is quite intuitive to understand that like for the Feynman Hamiltonian, the particular gates implemented do not inﬂuence the string dynamics. 5.3. Hamiltonian quantum computing on a lattice 97

We deﬁne the Hamiltonian Hvalid as

Hvalid = −∆ ∑ n[µ]n[ν] + (Ntrack − 1)∆I, (5.23) (µ,ν)∈E where ∆ > 0 and I the identity operator. Hvalid is clearly diagonal in the basis we have chosen and its spectrum can be directly obtained. The terms −∆n[µ]n[ν] at one edge can be interpreted as attractive interactions. In practice, any time a string will be disconnected at one point this will translate in an energy penalty equal to +∆, and consequenly the Hamiltonian Hvalid tends to favour conﬁgura- tions in which the strings are connected, meaning that they minimize the energy.

In fact, we immediately understand that the groundspace of Hvalid, restricted to the subspace with one excitation per track, is degenerate with eigenvalue E0 =

−(Ntrack − 1)∆ + (Ntrack − 1)∆ = 0, and it is composed of all the possible connected 2(N−1) strings. In particular, the number of possible connected strings is ( N−1 ) = NCN−1, where CN−1 is the (N − 1)-th Catalan number. This means that the groundspace of N Hvalid is 2 track NCN−1-dimensional, where we have also considered the internal degrees of freedom of course. The ﬁrst excited subspace is formed by the subspace with the strings disconnected at one point and consequently has an energy gap equal to

∆ compared to the groundspace, i.e., E1 = ∆. In general Hvalid separates in Ntrack − 2 eigensubspaces with energy Ek = k∆, where k ∈ {0, 1, . . . , Ntrack − 1} denotes the number of points at which the string is broken. Note that this Hamiltonian is fully degenerate with respect of the spin degree of freedom. We will see that in order to guarantee that the system remains in the groundspace we will need large gap ∆ compared to the other energy scale in the system.

5.3.4 Hopping Hamiltonian

We have already encountered the basic structure of the hopping Hamiltonian in Sub- sec. 5.3.1. In the case in which we want to run only single-qubit gates, we associate 2 to each plaquette a single-qubit unitary Up, with p ∈ PL for a total of (N − 1) single-qubit gates. The hopping Hamiltonian is deﬁned as

Vhop = −J ∑ Vhop,p, (5.24) p∈PL where Vhop,p is the hopping associated to the plaquette p and deﬁned as

1 1 0 † Vhop,p = ∑ ∑ hs | Up |si as0 [i + 1, j + 1]as[i, j] + h.c.. (5.25) s=0 s0=0

5.3.5 Effective Hamiltonian

The hopping Hamiltonian Vhop is not diagonal in the eigenbasis of Hvalid and can cause transitions from connected strings to connected strings, which in a quantum 98 Chapter 5. Hamiltonian quantum computing optics language are resonant, but also from a connected string to a disconnected string (off-resonant), which of course we would like to avoid. We are actually in a typical setting of perturbation theory in which H = Hvalid + Vhop with a Hamiltonian

Hvalid with known spectrum whose low lying eigenstates (the groundspace) is separated by a gap ∆ from the other excited states. In addition, we have a perturbation

Hamiltonian Vhop and we are interested in the low energy properties of this Hamil- tonian. In analogy to what we did in Chap.4 in the different setting a transmon or a TCQ coupled to two resonators, we can use a Schrieffer-Wolff transformation to study the problem. Let us denote by P− the projector onto the groundspace of

Hvalid, i.e., the subspace of connected strings. We also deﬁne P+ = 1 − P− the projectors onto the subspace of disconnected strings. As discussed in Refs. [119, 161] the ﬁrst order Schrieffer-Wolff consists just in the projection of the Hamiltonian onto the groundspace, and gives the effective Hamiltonian

2 2 Heff = H−− + O(kVhopk /∆) = (Vhop)−− + O(kVhopk /∆). (5.26)

The small parameter of the perturbation theory is kVhopk/∆, which can basically be connected to |J/∆| given a ﬁxed lattice size. In our problem we can give a simple interpretation to (Vhop)−−: it consists of all the hoppings that keep the string connected. In particular,

2 Heff = −J ∑ Hcond.hop,p + O(kVhopk /∆), (5.27) p∈PL where we deﬁned the conditional hopping Hamiltonian in a plaquette as

Hcond.hop,p = n[i + 1, j]n[i, j + 1]Vhop,p. (5.28)

The conditional hopping can take place in a plaquette only if there are particles at the top and the bottom sites of the plaquette, which guarantees that the string stays connected. As we see the initial Hamiltonian Eq. 5.22 is 2-local, where the deﬁnition of k- local Hamiltonian we are using is that of Def. 5.2.1, adapted to our case of hopping particles and not with qubits 5. Additionally, the Hamiltonian presents also a high degree of spatial locality, involving nearest- and next-nearest neighbour interactions. The effective Hamiltonian obtained perturbatively is instead 4-local. Assuming that the system starts in the conﬁguration in which all particles are on the left (see Fig. 5.3), the dynamics of the string under Heff in Eq. 5.27 can be exactly solved. In particular, this kind of dynamics has been analyzed in [158] via a Jordan-Wigner transformation and also in [160], where the motion, in the limit of N → +∞ was considered as a quantum walk on a graph. The idea of the Jordan- Wigner is related to the fact that the dynamics of the connected string can be related

5We will later show that we can map the problem to qubits and that the resulting Hamiltonian is indeed 2-local according to Def. 5.2.1. 5.3. Hamiltonian quantum computing on a lattice 99

...

∅

FIGURE 5.3: Quantum walk of the connected string. to the one-dimensional XY model and in particular to the sector of the XY model with N excitations. The key step of this mapping is to represent a connected string conﬁguration in the following way. Starting from the top vertex of the lattice and going towards the bottom vertex we write 0 if the string turns left, while 1 if the string turns right. As an example, the initial connected string in which all particles are on the left of the lattice would be represented by the ket

|00 . . . 0 11 . . . 1i . (5.29) | {z } | {z } N N

In this representation there must always be an equal number of zeros and ones since the string starts at the top vertex and ends at the bottom vertex. From this we realize that the only allowed transitions in this representation are those that exchange neighbouring zeros and ones, which is indeed the XY model restricted to the subspace with N particles. An equivalent way of interpreting the problem is to see it as a quantum walk on a restricted Young’s lattice, in the sense that each partition cannot exceed N. This is depicted in Fig. 5.3, where we recognize the first part of the Young’s lattice. If N is finite we will eventually hit the boundary, which causes the restriction of the Young’s lattice. In the limit of N → +∞ the dynamics becomes equivalent to a quantum walk on the infinite Young’s lattice, and it turns out that an analytic solution can be obtained using poset theory [160], without using the mapping to the XY model and the Jordan-Wigner transformation. Let us now describe how a quantum computation can be performed on this lattice. The following discussion is based on some rigorous results obtained in Refs. [158, 160]. At the beginning of the computation the excitations (one per track) are all placed on the left of the lattice, and, without loss of generality, in the |0i logical state. The quantum circuit is encoded in a K × K region of the lattice located in the left corner with K = N/4, where we are assuming that N is a multiple of 4. In the remaining part of the lattice only trivial identities are applied. We call this region the output region. If all particles are found in the output region, then the quantum 100 Chapter 5. Hamiltonian quantum computing computation has been completed, and we can measure each qubit in the computational basis to extract the result. As discussed in [160] (see in particular Theorem 3 of the Supplementary Material), by choosing the measurement time randomly and uniformly distributed in [0, T], with T = cN3 and c a positive constant, the prob- √ ability of finding all the particle in the output region is ≥ 1/4 + O(1/ N), which guarantees a polynomial scaling. We point out however that the ability to perform measurements on large areas of the lattice simultaneously is non-trivial in any practical implementation, and it is still unclear how this can be optimized and how it influences the dynamics, since a continuous search for the string may slow down the computation causing quantum Zeno effect [162].

5.4 Multi-qubit gates

So far we have restricted our attention to the case in which we have only quantum circuits involving single-qubit gates. If our purpose were to simulate these kinds of circuits the Hamiltonian Hvalid would be almost completely useless. In fact, as discussed in Subsec. 5.3.1, we could run each sequence of single-qubit gates in parallel on each track. In this setting the only reason why one would want to have a

Hvalid might be to make the dynamics more efﬁcient, but this does not seem worth the effort.

As we mentioned previously, the role of Hvalid is to coordinate the quantum circuit and ensure that the quantum logic is applied correctly, when we consider gates involving more than one qubit. Additionally, as it will be clear in the following sections, a modiﬁcation of Hvalid can allow the implementation of controlled and controlled-controlled unitaries, such as CNOT and Toffoli gates. We know that quantum universality can be achieved using Hadamard, T and CNOT gate [4]. However, we will show explicitly how to construct a Toffoli gate directly, so that universality can also be achieved with Hadamard and Toffoli gates [163, 164]. This turns out to be quite crucial in the implementation with superconducting qubits that we will describe in Sec.6, since we will be able to implement only real gates. While performing a Toffoli gate directly is clearly advantageous, universality with real gates can also be achieved with Hadamard, controlled-Hadamard and CNOT as we discuss in Ap- pendixB. Let us explain the basic idea behind a CNOT gate, which applies also to any controlled-U and generalizes to controlled-controlled operations. We will always consider a CNOT between adjacent tracks. The fact that in our scheme the string remains always connected guarantees that the target qubit does not pass the CNOT region if the control does not. This is however, insufﬁcient to implement a controlled gate. In fact, we should additionally require that the identity or the X gate is applied 5.4. Multi-qubit gates 101

(i, j − 1) (i + 1, j) Control (i, j, 0) (i − 1, j − 1) (i + 1, j + 1) X I Target I I (i, j, 1)

(i − 1, j) (i, j + 1)

FIGURE 5.4: The CNOT gate. to the target depending on the state of the control. The idea will be again to give energy penalties to these incorrect conﬁgurations. We explain this idea mathematically in the next subsection.

5.4.1 CNOT

We will now describe how a CNOT can be implemented following the construction in [19]. This will serve us as a warm up to understand the new construction of the Toffoli. Since we allow interactions only between particles on the same track and between particles on adjacent tracks, we are restricting ourselves to quantum circuits in which linear nearest-neighbour (LNN) two-qubit gates are allowed, also called one- dimensional quantum circuits. This is a common feature also of many practical implementations of the circuit model of quantum computation and the consequences of these limitations have been already analyzed in the literature [165, 166, 167]. For us it suffices to say that any quantum algorithm designed in the standard circuit-model can be simulated in the restricted LNN model with at most a polynomial overhead. In order to describe the CNOT let us consider Fig. 5.4. Compared to the construction with only single-qubit gates we need to modify a region of the lattice. In particular, the central site has to be split into two sites and depending on the internal state of the control particle, the target particle is directed to one of the two intermediate sites. An X gate or the identity gate I is applied depending on whether the particle hops to the site associated with state |1i of the control or state |0i. The CNOT is finally completed with the hopping of both intermediate sites to a common final site. This way of implementing a CNOT is reminescent of a railroad switch that is then brought back together on a single track. We remark that a similar idea can be found in Ref. [148]. As we have already mentioned, the idea to ensure that the logic is implemented correctly is to give energy penalties to incorrect configurations. By incorrect configurations we here mean for instance the case in which the control is in the logical |0i, but the target ended up in the site associated to logical state |1i and so an X gate has been applied incorrectly. Referring to Fig. 5.4 we define annihilation, creation and 102 Chapter 5. Hamiltonian quantum computing number operators for the two sites in the center

† ns[i, j, k] = as [i, j, k]as[i, j, k], (5.30)

1 1 n[i, j, k] = ∑ ns[i, j, k], n[i, j] = ∑ n[i, j, k] (5.31) s=0 k=0 . For simplicity of notation we will consider the case in which we have only one

CNOT gate in our quantum circuit. We need to modify the deﬁnition of Hvalid given in Eq. 5.23, and speciﬁcally the terms associated with the green edges a, b, c, d depicted in Fig. 5.4. We denote the set of these edges as ECNOT The valid Hamiltonian becomes

Hvalid = −∆ ∑ n[µ]n[ν] − ∆ ∑ He + (Ntrack − 1)∆I. (5.32) (µ,ν)∈E\ECNOT e∈ECNOT

The Hamiltonians associated with the edges e ∈ ECNOT are modiﬁed in the following way. Edges c and d preserve the same structural form as for the the single-qubit case

Hc =n(i − 1, j)n(i, j), (5.33a)

Hd =n(i, j + 1)n(i, j) (5.33b) the only difference is in the deﬁnition of n(i, j).Using the deﬁnitions for the number operators given in Eqs. 5.30, 5.31, the Hamiltonians associated with edges a and b are given by

Ha = ns=0[i, j − 1]n[i, j, k = 0] + ns=1[i, j − 1]n[i, j, k = 1], (5.34a)

Hb = ns=0[i + 1, j]n[i, j, k = 0] + ns=1[i + 1, j]n[i, j, k = 1]. (5.34b)

These last two terms give energy penalties to incorrect conﬁgurations in the same way as the normal valid Hamiltonian gives penalties to disconnected strings. To see this let us consider a connected test string that passes through edge a and let us analyze its contribution. Since we know that this string is connected, all the other edges give a total energy contribution equal to ∆. The contribution of edge a is also readily analyzed. We see that if the control is in the s = 0 state and the target is located in the center site with label k = 0, or if the control is s = 1 and the target is in the site with label k = 1, edge a gives a contribution −∆. In this case the CNOT logic is implemented correctly. In fact, if the target ended up in site (i, j, 0), this would mean that the identity has been applied, which is the correct logical operation if the control is in the logical |0i state. The same reasoning applies to the case in which the target particle ends up in (i, j, 1). We conclude that a valid string, that is connected and correct, has eigenvalue of Hvalid equal to zero. Instead, if the logic is not implemented correctly, edge a gives zero contribution, and thus a connected, 5.4. Multi-qubit gates 103 but incorrect string has energy ∆. The same analysis applies to edge b. From this reasoning we understand that the groundspace of the modiﬁed Hvalid Hamiltonian in Eq. 5.32 is not formed simply by the connected strings, but by valid strings, which are connected and correct. We still have to complete the description of the CNOT by specifying the form of the hopping Hamiltonian. We have to modify the contribution to the hopping

Hamiltonian of plaquettes p1 and p2 in Fig. 5.4. We call the set of these plaquettes

PLCNOT. The hopping Hamiltonian in Eq. 5.24 in presence of a single CNOT is modiﬁed as Vhop = −J ∑ Vhop,p − J ∑ Vhop,p. (5.35) p∈PL\PLCNOT p∈PLCNOT

The second sum involves two terms Vhop,p1 and Vhop,p2 , identiﬁed by the green lines in Fig. 5.4. Vhop,p1 applies either the identity or the X gate depending on whether the particle is hopping to (i, j, 0) or to (i, j, 1)

1 1 V = a†[i j ]a [i − j − ] + + a†[i j ]a [i − j − ] + hop,p1 ∑ s , , 0 s 1, 1 h.c. ∑ s¯ , , 0 s 1, 1 h.c., (5.36) s=0 s=0

where we denoted as s¯ the negation of s. Vhop,p2 ﬁnally hops the particle in the intermediate sites to the ﬁnal common site in (i + 1, j + 1)

1 1 = †[ + + ] [ ] + Vhop,p2 ∑ ∑ as i 1, j 1 as i, j, k h.c.. (5.37) s=0 k=0

As a ﬁnal remark we point out that the control qubit needs to undergo the identity when hopping from site (i, j − 1) to site (i + 1, j).

The groundspace of Hvalid Eq. 5.32 is now composed of all valid strings. We also notice that the first excited subspace is composed of all the strings broken at one position and of connected, but incorrect strings. The energy of the first excited subspace is still ∆ so the gap does not change. By defining P− the projector onto the new groundspace, the effective Hamiltonian is again in first order Schrieffer-Wolff given by H−−

2 Heff = −J ∑ Hcond.hop,p − J ∑ Hcond.hop,p + O(kVhopk /∆), (5.38) p∈PL\PLCNOT p∈PLCNOT where the terms in the ﬁrst sum involve only single-qubit gates and they are deﬁned as in Eq. 5.28. The second sum gives instead two terms

1 H = n [i j − ]n[i − j] a†[i j ]a [i − j − ] + + cond.hop,p1 s=0 , 1 1, ∑ s , , 0 s 1, 1 h.c. s=0 1 † ns=1[i, j − 1]n[i − 1, j] ∑ as¯ [i, j, 0]as[i − 1, j − 1] + h.c. , (5.39) s=0 104 Chapter 5. Hamiltonian quantum computing

1 1 = [ + ]n[ + ] †[ + + ] [ ] + Hcond.hop,p2 ∑ nk i 1, j i, j 1 ∑ as i 1, j 1 as i, j, k h.c. . (5.40) k=0 s=0

(i, j − 1) (i + 1, j) (i + 2, j + 1) Control 1 a1 b a2 b (i, j, 0) 1 (i + 1, j + 1, 0) 2 I (i − 1, j − 1) I I I (i + 2, j + 2) Target p1 p3 p I 2 I I X (i, j, 1) (i + 1, j + 1, 1) c1 d1 c2 d2 Control 2 (i − 1, j) (i, j + 1) (i + 1, j + 2)

FIGURE 5.5: Direct Toffoli gate.

5.4.2 Toffoli gate

The construction of the Toffoli gate closely resembles the one of the CNOT, where we now have to consider two control particles. Like for the CNOT we will be able to consider quantum circuits in which we can implement only local Toffoli gates directly. As shown in Fig. 5.5, we consider only Toffoli gates in which the target is sandwiched between the two control qubits. Again any quantum algorithm involving long range Toffoli gates can be mapped to one with local Toffoli gates with a polynomial overhead. From Fig. 5.5 we see that in order to implement a Toffoli gate we need to modify a larger region compared to the CNOT. We now need to split two sites. Another possibility would have been to split a single site in four. We have however discarded this option, because in addition it would have required an interaction between the two control qubits. The idea of the Toffoli is quite simple once we know how to perform the CNOT. In the first plaquette we just apply the first step of the CNOT with the particle on the Control 1 track as control. There is however one difference: the X gate is not immediately applied, but the particle is just directed to the site (i, j, 0) or (i, j, 1) according to the state of the control. After this intermediate step, the target particle is subjected to conditional operations depending on the state of the second particle on the Control 2 track. In order for this to be possible, the control needs to be in site (i, j + 1). As an example if the target is in (i, j, 1) and the second control is the s = 1 state the target particle will be allowed to hop only to the site (i + 1, j + 1, 1) with application of an X gate correctly realizing the logic of the Toffoli gate. The system works similarly in the remaining three cases in which eventually the identity is applied. Finally, from the second intermediate sites (i + 1, j + 1, 0) and (i + 1, j + 1, 0) the target hops to the final single site in (i + 2, j + 2) with the application of the identity. This construction of the Toffoli can be viewed as a double, sequential railroad switch. 5.4. Multi-qubit gates 105

In order to describe these ideas mathematically, like for the CNOT case, we consider, for the sake of simplicity, only quantum circuits with one Toffoli gate. We need to modify the valid Hamiltonian associated to the red edges in Fig. 5.5. The notation is extended naturally from that introduced for the CNOT case. We denote the set of these edges ETOF. We write Hvalid as

Hvalid = −∆ ∑ n[µ]n[ν] − ∆ ∑ He + (Ntrack − 1)∆I. (5.41) (µ,ν)∈E\ETOF e∈ETOF

The valid Hamiltonian associated to edges c1, d1 and a2, b2 does not change compared to the case with single-qubit gates. Hence we write

Hc1 =n(i − 1, j)n(i, j), (5.42a)

Hd1 =n(i, j + 1)n(i, j), (5.42b)

Ha2 =n(i + 1, j)n(i + 1, j + 1), (5.42c)

Hb2 =n(i + 2, j + 1)n(i + 1, j + 1). (5.42d)

The remaining edges in ETOF are modiﬁed as

Ha1 = ns1=0[i, j − 1]n[i, j, k = 0] + ns1=1[i, j − 1]n[i, j, k = 1], (5.43a)

Hb1 = ns1=0[i + 1, j]n[i, j, k = 0] + ns1=1[i + 1, j]n[i, j, k = 1]. (5.43b)

Hc2 = ns2=0[i, j + 1]n[i + 1, j + 1, k = 0] + ns2=1[i, j + 1]n[i + 1, j + 1, k = 1], (5.43c)

Hd2 = ns2=0[i + 1, j + 2]n[i + 1, j + 1, k = 0] + ns2=1[i + 1, j + 2]n[i + 1, j + 1, k = 1]. (5.43d) Again in this construction, like for the case of a CNOT, one can check that valid strings have zero energy, while the gap is unchanged and equal to ∆. We now need to modify the hopping Hamiltonian associated with plaquettes p1, p2, p3 in Fig.5.5.

We denote the set of these plaquettes as PLTOF and write

Vhop = −J ∑ Vhop,p − J ∑ Vhop,p, (5.44) p∈PL\PLTOF p∈PLTOF where the three terms in the last sum are given by

1 1 V = a† [i, j, k]a [i − 1, j − 1] + h.c. (5.45a) hop,p1 ∑ ∑ s1 s1 s1=0 k=0 106 Chapter 5. Hamiltonian quantum computing

1 1 V = a† [i + 1, j + 1, 0]a [i, j, 0] + h.c. + a† [i + 1, j + 1, 0]a [i, j, 1] + h.c.+ hop,p2 ∑ s2 s2 ∑ s2 s2 s2=0 s2=0 1 1 a† [i + 1, j + 1, 1]a [i, j, 0] + h.c. + a† [i + 1, j + 1, 1]a [i, j, 1] + h.c. (5.45b) ∑ s2 s2 ∑ s¯2 s2 s2=0 s2=0

1 1 V = a† [i + 2, j + 2]a [i + 1, j + 1, k] + h.c.. (5.45c) hop,p3 ∑ ∑ s2 s1 s2=0 k=0 In order to obtain the new effective Hamiltonian we just need to project as usual onto the new groundspace of connected and correct strings. In this case correct means implementing the correct Toffoli logic. We can write the effective Hamiltonian as

2 Heff = −J ∑ Hcond.hop,p − J ∑ Hcond.hop,p + O(kVhopk /∆), (5.46) p∈PL\PLTOF p∈PLTOF where the second sum involves the conditional hopping terms related to plaquette

p1, p2, p3

1 1 H = n [i j − ]n[i − j] a†[i j s ]a [i − j − ] + cond.hop,p1 ∑ ∑ s1 , 1 1, s , , 1 s 1, 1 h.c. , (5.47a) s1=0 s=0

1 1 H = n [i, j + 1]n [i + 1, j] a† [i + 1, j + 1, s ]a [i, j, s ] + h.c. , cond.hop,p2 ∑ ∑ s2 s1 s⊕s1s2 2 s 1 s1,s2=0 s=0 (5.47b)

1 1 = [ + + ]n[ + + ] †[ + + ] [ + + ] + Hcond.hop,p3 ∑ ∑ ns2 i 1, j 2 i 2, j 1 as i 2, j 2 as i 1, j 1, s2 h.c. . s2=0 s=0 (5.47c) We conclude this section with some comments about the implementation of the Tof- foli gate that we have just presented. It is clear that the same construction can be used to implement any controlled-controlled-U gate just by replacing the hopping term that implements the X gate with a hopping term that implements a generic single-qubit unitary as described in Subsec. 5.3.4. In addition, one might wonder if all the hopping terms described here are really necessary. Let us consider again

Fig. 5.5 and let us suppose that the first control qubit is in the s1 = 0 state. The target qubit can then hop to the site (i, j, 0), assuming our perturbative Hamiltonian to be exact. At this point we are tempted to say that no matter what the state of the second control is, the identity has to be applied and so the hopping from (i, j, 0) to (i + 1, j + 1, 1) looks useless. This is however not the case, and the reason why we are led to this mistake is that we selected a particular direction of the computation, ı.e., from left to right, while terms that hop in the opposite direction are always present by the hermiticity of the Hamiltonian, which ensures that everything has 5.5. Back to Feynman 107 to be reversible. In fact, in order to guarantee that the position of the particles always identifies univocally the stage of the computation it is necessary that when the string runs backwards the computation is undone. This would not be the case if we removed the identity connecting (i, j, 0) and (i + 1, j + 1, 1) and we can understand this by trying to run the circuit backwards and showing that it cannot implement a Toffoli gate. Suppose that the target particle is in (i + 2, j + 2) and the second control is in the state s2 = 1. If the target particle runs backwards, it would be directed to site (i + 1, j + 1, 1). However, at this point even if the first control qubit is in s1 = 0 the target can never go further backwards with the application of the identity since the hopping from (i + 1, j + 1, 1) to (i, j, 0) is missing. We conclude that if the systems hops backwards the Toffoli gate is not applied. For this reason, the interactions depicted in Fig. 5.5 and described mathematically in this section are all necessary. As a final remark, and in analogy with the CNOT, the control qubits must obviously undergo the identity in the Toffoli region.

5.5 Back to Feynman

In the previous two sections we have described the model for Hamiltonian quantum computing with hopping particles introduced by Ref. [19] and also explained how we can implement controlled-controlled-U gates in the model. In this section we introduce a closely related model, which differs compared to the previous one by the dynamics that the computation undergoes. We will show that this model maps unitarily to the Feynman Hamiltonian discussed in Sec. 5.2 when we project the Hamiltonian onto the groundspace. Consequently, the dynamics that the computation undergoes is equivalent to a quantum walk on one line, with a length that is not given by the number of gates applied since we need to count the two-qubit gates twice and the three-qubit gates three times. An advantage of this scheme is that the dynamics is easier to analyze and we can apply well known ideas of perfect quantum state transfer to guarantee the existance of a time for which the computation is completed with deterministic time [145]. This however cannot be practically done for an arbitrary large computation since it will require an unphysical increase of the coupling strength. Another advantage of this construction is that all the qubits can be potentially used for the computation, in the sense that we do not need to limit the computational particles to be restrited to a certain area of the lattice. Additionally, the ability to measure a computational region simultaneously is non-trivial and it is so far unclear what is the effect of a certain measurement procedure on the dynamics. However, as an effective one-dimensional system, and in the presence of disorder, this model is affected by the problem of Anderson localization [168]. Before introducing the new model, let us describe the kind of quantum circuit we are going to implement. Like in Sec. 5.3 we ﬁrst focus for simplicity on the case in which only single-qubit gates are present and then extend the analysis to circuits 108 Chapter 5. Hamiltonian quantum computing

U11 U12 U11 U12 U11 U12 U11 U12 U11 U12 U11 U12

U21 U22 U21 U22 U21 U22 U21 U22 U21 U22 U21 U22 ...

U31 U32 U31 U32 U31 U32 U31 U32 U31 U32 U31 U32

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5

FIGURE 5.6: Example of depth 2 quantum circuit with 3 qubits and gates executed in a snake-like order. with two- and three-qubit gates. The model we will introduce will implement quantum circuits in which the gates are implemented in the order depicted in Fig. 5.6. Odd columns of the circuit are executed from top to bottom, while even columns from bottom to top. This has to be interpreted in a quantum mechanical language when we talk about quantum computing with a time-independent Hamiltonian. In a certain sense, the motion of the computation is reminiscent of that of a snake that runs through the quantum circuit in a construction similar to the one analyzed in Ref. [152]. The reason why we consider these kinds of quantum circuits is that we will manage to keep the interactions as spatially local as possible. We found that a related idea was presented in [169]. An immediate negative consequence is however that we are ruling out the possibility of executing some parts of the quantum circuit in parallel. This last possibility is also only partially present in the scheme described in Sec. 5.3. We consider the lattice in Fig. 5.7.It consists of N × M sites where the particles can reside. Also in this model, we will always assume that there is always one particle per track. We will show that the model we construct is able to encode a quantum circuit with N-qubits, where each qubit undergoes M − 1 unitaries. The blue edges represent the attractive interactions whose purpose is to keep the string together (in ﬁrst order perturbation theory). The orange edges instead will be associated with the hopping Hamiltonians that apply gates to the qubit degree of freedom. We denote the set of blue edges as Eb and the set of orange edges as Ey. A lattice point is speciﬁed by (i, j) with i ∈ 1, 2, . . . , N, j ∈ 1, 2, . . . , M 6. A generic unitary of our quantum circuit applied at a orange edge is denoted as Uij with i ∈ 1, 2, . . . , N, j ∈ 1, 2, . . . , M − 1. Annihilation, creation and number operators are introduced in analogy with Sec. 5.3. The Hamiltonian can again be written as the sum of two contributions H = Hvalid + Vhop as in Eq. 5.22 with the valid Hamiltonian

0 Hvalid = −∆ ∑ n[ν]n[ν ] + (N − 1)∆I. (5.48) eb∈Eb

The groundspace of the valid Hamiltonian has again zero eigenvalue and it is composed of all the possible strings of particles that can be connected by the blue edges. The number of connected strings in this sense is L = N × (M − 1) + 1. The ﬁrst

6Notice that now N corresponds to the number of tracks. 5.5. Back to Feynman 109

j j

i i

(A) Connected string. (B) Disconnected string.

FIGURE 5.7: Examples of connected and disconnected strings of particles in the new lattice model. The red dots denote the position of the particles.

...

|0i |1i |2i |3i |4i |5i |6i

FIGURE 5.8: String motion. excited subspace instead is given by the subspace of strings that fail to be connected by blue edges at one position, and they have energy ∆, which corresponds to the gap. In what follows we will talk about connected and broken strings in the sense just described. An example of connected and disconnected string in this new model is given in Eq. 5.7. The hopping Hamiltonian is

n m−1 Vhop = −J ∑ ∑ Vhop,ij (5.49) i=1 j=1 with 1 1 0 † Vhop,ij = ∑ ∑ hs | Uij |si as0 [i, j + 1]as[i, j] + h.c.. (5.50) s=0 s0=0

By projecting the Hamiltonian onto the groundspace of connected strings we obtain again our effective Hamiltonian. Before showing explicitly the derivation, we now explain intuitively why the motion maps to a quantum walk on a line with L sites, following the order shown in Fig. 5.8. Referring to Fig. 5.8, we see that the system is initialized in a state in which all the particles are on the left, which is a connected configuration according to our definition. We can denote this configuration by |0i. There is only one move that starting from this configuration gives a different connected configuration, ı.e., the particle on the top can hop from site (1, 1) to site (1, 2) bringing the string to state |1i. Now there are two moves that we can do to keep the string connected. Either we undo the previous step and go back to |0i, or the particle in (2, 1) hops to (2, 2) leading to configuration |2i. We can proceed in this way till we reach the bottom of the lattice. In fact, if the system is in configuration |4i it can hop forward to configuration |5i and from |5i again upwards towards |6i 110 Chapter 5. Hamiltonian quantum computing and so forth until we reach the top again and the system starts to run downwards. We understand that this is exactly the order in which the gates are implemented in the kind of quantum circuit we are considering. Let us obtain this intuitive result mathematically. We associate with each connected string an integer k ∈ {0, 1, 2, . . . , L − 1}, via a bijective function f : S → {0, 1, 2, . . . , L − 1}, where we denoted by S the set of connected strings. A generic connected string is specified by the N points it connects, so that we can write s ∈ S as s = {(is1, js1),..., (isN, jsN)}. We denote the maximum column index j in a string as js,max, while the minimum js,min. The integer k = f (s) associated with the string is as  n(js,max − 1), js,min = js,max,  k = f (s) = ( − ) + N = 6= (5.51) n js,max 1 ∑l=1 δjsl,js,max , js,max 2l , js,max js,min,   ( ) − − N = 6= n js,max 1 ∑l=1 δjsl,js,min , js,min 2l , js,max js,min, with l an integer. From this mapping we can identify a connected string by its index k. The projector onto the connected string (valid) subspace is given by the sum of L−1) the single projectors onto a connected string P− = ∑k=0 Pk, where in terms of the number operators each projector Pk is given by

Pk = ∏ n[ν]. (5.52) ν∈s

The previous equation has to be intended to be restricted to the one excitation per track. Within this subspace the Pk are correctly projectors since they satisfy PkPk0 =

δkk0 . Thus, we can write

L−1 L−1 L−2 Heff = P− HP− = ∑ Pk Vhop ∑ Pk0 = ∑ PkVhopPk+1 + h.c.. (5.53) k=0 k0=0 k=0

In order to complete the mapping to the Feynman Hamiltonian we now deﬁne a convenient basis to analyze the problem. From the discussion in Subsec.5.3.2 we know ⊗N that our problem can be represented in a Hilbert space H = (HQ ⊗ Hclock) = ⊗N ⊗N (HQ ) ⊗ (Hclock), where each HQ ⊗ Hclock represents the equivalent system of a track, and each clock is M + 1 dimensional. The global clock has associated Hilbert ⊗N space Hclock. The projection onto the subspace of connected strings gives, in the global clock space, only conﬁgurations that we can identify with an integer k, which in this Hilbert space we associate with a ket |kic. In particular, projecting the Hamil- tonian we obtain matrix elements only between clock states |ki and its neighbours |k ± 1i. In this new Hilbert space we can then write Eq. 5.53 as

L−1 Heff = HF = −J ∑ Uk ⊗ |ki hk + 1|c + h.c., (5.54) k=0 which has the form of the Feynman Hamiltonian Eq. 5.1, where only single-qubit 5.5. Back to Feynman 111 gates are applied. Also in this case the role of the clock is played by the conﬁguration of the particles. We have shown in Sec. 5.2 how the dynamics of the Feynman Hamiltonian can be mapped to that of a quantum walk on one line, which in our case has L sites. This Hamiltonian reads

L−1 HL = −J ∑ |ki hk + 1| + h.c., (5.55) k=1 where we now denoted by |ki the ket associated to the k-th site. What we want to obtain is the probability as a function of time of ﬁnding the system in state |Li when it starts at time t = 0 in state |1i, namely

2 −iHLt 2 PL(t) = |F(t)| = |hL| e |1i| . (5.56)

The Hamiltonian HL has eigenkets r 2 L πlk |l˜i = sin |ki , (5.57) + ∑ + L 1 k=1 L 1 and corresponding eigenvalues

πl ω˜ = −2J cos , (5.58) l L + 1 with l = 1, . . . , L. The transition amplitude F(t) can be written as

1.0 L = 20 L = 100 0.8

0.6 L P 0.4

0.2

0.0 0 20 40 60 80 100 Jt

FIGURE 5.9: Success probability PL(t) for L = 20 and L = 100 as a function of Jt. 112 Chapter 5. Hamiltonian quantum computing and L = 100. We see that although the maximum probability is still quite high at L = 100, it deﬁnitely decreases compared to the case L = 20 and in general the same is true increasing L. For instance the maximum probability with L = 1000 is approximately 0.1, which is however not negligible. Additionally, the time at which the maximum probability is reached scales linearly with the number of sites, ı.e., in our original model, with the number of gates applied.

5.5.1 Peres’ trick

In this subsection, we just apply a well known idea of perfect quantum state transfer originally due to Peres [146] exactly in his analysis of the Feynman Hamiltonian, and rediscovered in [144], in the context of perfect quantum state transfer. It is also argued that this way of achieving perfect state transfer is optimal from several points of view [145]. We have seen that after the projection onto the groundspace, the dynamics of our model is equivalent to that of a quantum walk on the line Eq. 5.55. We have assumed that the parameter J is the same for all the hoppings, but this is not a necessary condition. In the general case in which J is allowed to vary the dynamics reduces to L−1 HL,d = − ∑ Jk |ki hk + 1| + h.c., (5.60) k=1

The idea is now to engineer the Jk in such a way that the Hamiltonian is proportional to the angular momentum operator Jx of a ﬁctitious spin-(L − 1)/2 particle. The basis states on the line |ki will be interpreted as the eigenkets of the operator J 2 and

Jz. To this end we relabel the basis states as

|ki → |m = −(L − 1)/2 + k − 1i , k ∈ {1, . . . , L}. (5.61)

2 We need the matrix elements of the operators J+ and J− in the basis of J , Jz. The eigenvalue of J 2 is ﬁxed and equal to (L − 1)/2, so we can just label the eigenkets of Jz with the index m (m = −(L − 1)/2, −(L − 1)/2 + 1, . . . , (L − 1)/2). The matrix elements are given by (see Chap 3. of [27])

s 0 L − 1 L − 1 hm | J± |mi = ∓ m ± m + 1 δ 0 ± , (5.62) 2 2 m ,m 1 from which we also directly obtain the matrix elements of Jx = (J+ + J−)/2. In order to obtain a Hamiltonian proportional to Jx we thus set (in the initial labeling of states) L−1 HL,d = −J ∑ λk |k + 1i hk| + h.c. = −JJx, (5.63) k=1 with 1q λ = k(L − k). (5.64) k 2 5.5. Back to Feynman 113

At this point the quickest way to show that there is a time T such that a system starting in the state |1i ends up in the state |Li with unit probability, under the action of U(t) = exp[iJx Jt], is to remember that this operator performs a rotation around the x-axis by an angle φ = −Jt. Let us denote by Dx(φ) = exp[−iJxφ] the operator associated with a rotation around the x-axis by an angle φ in a generic L-dimensional system. Using the angular momentum commutation relations

[Ji, Jj] = iεijkJk (5.65) and the identity

1 1 eABe−A = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . . (5.66) 2! 3! valid for general operators A and B, we get the action of Dx(φ) on the angular momentum operators Jx, Jy, Jz

† Dx(φ) JxDx(φ) = Jx, (5.67a)

† Dx(φ) JyDx(φ) = cos(φ)Jy − sin(φ)Jz, (5.67b)

† Dx(φ) JzDx(φ) = cos(φ)Jz + sin(φ)Jy. (5.67c)

The previous equations are exactly how we would expect a vector in three dimensions to transform under a rotation around the x-axis by an angle φ. We can use this intuition to understand when, and if, the system reaches the ﬁnal state |ψﬁni = |Li with probability 1, starting from |ψini = |1i. The initial state is an eigenket of Jz with minimum eigenvalue −(L − 1)/2. Consequently, the average of Jz in this state is L − 1 hJ i = − . (5.68) z in 2

The ﬁnal state instead is also eigenket of Jz with maximum eigenvalue (L − 1)/2 and thus

hJzi f = − hJziin . (5.69)

Using Eq. 5.67c

hJ i ( ) = h | −iJx JtJ iJx Jt | i = ( ) hJ i − ( ) hJ i z t ψin e ze ψin cos Jt z in sin Jt y in . (5.70)

hJ i = = = Notice that in the previous equation y in 0. At time t T π/J we have the desired condition hJzi (T) = − hJziin, which is easy to understand implies that the system is found in |ψﬁni = |Li with probability 1. This is due to the fact that |Li is eigenket with maximum eigenvalue (L − 1)/2 of Jz, so if the average is (L − 1)/2 necessarily |ψi (T) = |Li. 114 Chapter 5. Hamiltonian quantum computing

It is actually convenient to rewrite the Hamiltonian explicitly in terms of the hitting time T as

π L−1 π HL,d = − ∑ λk(|ki hk + 1| + |k + 1i hk|) = − Jx. (5.71) T k=1 T

The result we have just obtained is quite remarkable. In the next section we explain how to extend it to the case in which we also have two-qubit gates with an analogous result. It actually seems that one can do an arbitrary quantum computation in a deterministic time T = π/J, which is independent of the number of gates applied. This is only apparently the case. In fact, the scheme requires that the coupling parameters Jk scale with the number of gates, and the allowed variation of these parameters has always some limitations due to the particular experimental realization we consider. However, the scaling is quite advantageous. For illustration purposes we assume the total number of applied gates L, ı.e., the steps in the quantum walk, to be even. Suppose we fix the maximum allowed hopping parameter to be less or equal a certain Jmax. In the previous scheme the maximum coupling is located in the middle and it is given by JL/2 = J/4L = Jmax, which we set equal to our Jmax to saturate the inequality. This means that if we keep the maximum Jk fixed, then the computational time will be T = π4L/Jmax, and thus scales linearly with L. This is indeed approximately our case, in which the Jmax will be limited by the achievable strength of the attractive interactions ∆. There is an additional subtlety though. The previous argument can be applied to our scheme strictly speaking if we keep the number of qubits N fixed in this discussion. We will explain this better in a Sec.

5.7, but the point is that the maximum allowed Jk depends on the number of qubits (tracks), but not on the number of columns.

5.5.2 Multi-qubit gates in the new model

The construction of the CNOT and Toffoli gates, and in general of any controlled and controlled-controlled unitary is, from a mathematical point of view, very similar to the one of the original model discussed in Sec.5.4. There are however some subtleties that should be clarified. In particular, we will show that there are two ways of approaching the problem depicted in Fig. 5.10. In Fig. 5.10a we represented what we called the slow CNOT. In this case the Hamiltonian of four blue edges for every CNOT needs to be modified, compared to the case in which we have only single-qubit gates. This is the same situation as the Lloyd-Terhal scheme. The CNOT begins in one column, but it is terminated in the next one. In the meantime all the gates below the target in the first column and in the second column need to be executed before the CNOT can be finally terminated. This means that the control and the target are busy for two rounds of the computation and not only one, and cannot be involved in other gates. To avoid this problem, we might consider to implement the CNOT as in Fig. 5.10b, which we called a fast CNOT since it can be 5.5. Back to Feynman 115

. . Control

X I Target I I

. .

(A) Slow CNOT. . . Control

I Target X I I

. .

(B) Fast CNOT

FIGURE 5.10: CNOT in the ‘single-clock snake’ lattice model of Fig. 5.7. The green edges are those who need to be modiﬁed similar to what we do in Subsec. 5.4.1. accomplished in a single round. The idea is to add the split site of the CNOT in the middle of a hopping edge, and also add the necessary attractive edges (the green ones in the picture). We can immediately check that if the string needs to stay connected, then the CNOT is fully executed and afterwards the computation continues to run downwards in the particular case of Fig. 5.10b. We point out that the control qubit can be either above or below the target in any case. This second construction is advantageous because we can immediately say that each CNOT would contribute two sites in the quantum walk on the line, while this conclusion cannot be drawn for the slow CNOT, in which case the effect would be always algorithm dependent. However, the fast CNOT has the disadvantage that it requires a region more dense in interactions. An analogous discussion can be given for the Toffoli. 116 Chapter 5. Hamiltonian quantum computing

∆1

∆2

(A) Allowed variation of the attractive interactions.

Ω1

Ω2

Ω3 (B) Allowed variation of the on-site energy.

FIGURE 5.11: Freedom in the realization of the model. In (A) different colors represent different attractive interactions, while in (B) different colors denote different on-site energy.

5.6 Some freedom in the model

In the models we have introduced in this chapter we required the parameter ∆ to be equal for all attractive edges. This is not a necessary condition though. In fact, the important thing is that the parameter ∆ is the same for all edges between two adjacent tracks, but it can be different when we consider different pairs of adjacent tracks without affecting the model itself. This is shown for a small lattice portion in

Fig. 5.11a. If we allow different ∆k the gap will then be min(∆k). Moreover, due to the fact that we always have one excitation per track we can add to all the sites on a certain track T local on-site terms ΩTn[ν] without any effect. The particular ΩT might however be different between different tracks, as depicted in Fig. 5.11b. Al- though these local ﬁelds do not inﬂuence the ideal model, they have an effect when we consider a physical implementation. For instance we know that at the edges we need attractive interactions. In practice, in most cases, these kinds of interactions will always come together with some other terms, which typically can cause hopping between different tracks. Having detuned lines helps in mitigating the effect of these terms.

5.7 Analysis of unitary errors

In this section we analyze several sources of unitary errors in our models, focusing mainly on the Lloyd-Terhal scheme described in Sec. 5.3.2. 5.7. Analysis of unitary errors 117

J/ = 1/10

0.12 N = 3 N = 4 0.100 0.10 N = 5

s 0.075 s

0.08 , D

P 0.050 D 0.06 P 0.025 0.04 3 4 5 N 0.02

0.00 0 1 2 3 4 5 6 7 8 9 10 Jt

FIGURE 5.12: Time averaged probability for the strings to be disconnected PD as a function of time for different lattice sizes. The inset shows the scaling of the steady state probability PD,ss (evaluated at the ﬁnal time) with N.

5.7.1 Perturbation theory analysis

In this subsection we analyze errors in the model, originating from the fact that our effective Hamiltonian is arrived at in lowest-order perturbation theory, as discussed in Subsec. 5.3.5. The features we are going to discuss are common to both models we analyzed in this chapter. First of all, we need to understand how the parameters should scale with the size of the problem. We know that in order to be in the regime of validity of our perturbation theory we need kVk/∆ 1. It is clear that the norm of the perturbation V increases with the size of the lattice. However, we here show that it does not scale with the total number of sites, but only with the total number of tracks. This is a consequence of the fact that we assume all our Hamiltonians to be restricted to the subspace with one excitation per track. In fact, in both models, we can always write our perturbation as

V = ∑ VQW,l, (5.72) l where the sum runs over the horizontal lines and VQW,l is the single-particle quantum walk associated with the track l, whose form is given in Eq. 5.55. We assume for concreteness equal hopping coefficient J, but similar conclusions can be drawn if J varies. We then have kVk ≤ ∑kVQW,lk = 2NT|J|, (5.73) l where the last equality follows from Eq. 5.58, and NT is the number of tracks. Thus, importantly, given a fixed number of implemented qubits, we can increase the number of implemented gates without making our approximations worse. Increasing the number of implemented qubits, ı.e., the number of tracks, and keeping fixed the ratio J/∆, we can expect the probability for the string to be disconnected to scale at most linearly with the lattice size from the previous argument. We have confirmed 118 Chapter 5. Hamiltonian quantum computing

J/ = 1/50 5.0 N = 3 4.5 N = 4

4.0 N = 5

3.5

X 3.0

2.5

2.0

1.5

1.0 0 1 2 3 4 5 6 7 8 9 10 Jt

FIGURE 5.13: Average position of the particle on the central track as a function of time for different lattice sizes N. The dashed horizontal lines identify the maximum value that can be reached for each lattice size. this numerically for small lattice sizes as shown in Fig.5.12, where we plot the time- averaged probability for the string to be disconnected

Z t 1 0 0 PD(t) = dt PD(t ), (5.74) t 0 as a function of time for different lattice sizes, ﬁxing J/∆ = 1/10. We see that unlike the instantaneous probability, the time-averaged probability reaches a steady value after a time of few 1/J. In the inset we see that this steady-state average probability

PD,ss scales linearly with the lattice size for these small lattice sizes.

5.7.2 Wavefront analysis and effect of disorder

We further analyze the dynamics of the string of particles for the Lloyd -Terhal model in Fig. 5.13. In particular, we plot the instantaneous average position of the particle on the central track, which is representative of the speed of the string. We see that at short times of order 1/J the particle moves at a constant velocity which is approximately ≈ 0.6[lattice sites]/(Jt) and independent of the lattice size, at least for the lattice sizes we studied. This agrees with the analysis of Ref. [160] in which, based on the solution for N → +∞, it is argued that the string should move at constant velocity until it reaches the boundaries. In Fig. 5.14 we further show some screenshots of the wavefront for lattice size N = 5 and at different times. We see that the particles tend to bunch together, as expected, and move forward in a correlated way. Another important issue that we have to take into account is how imperfections, which will be present in any practical implementation, inﬂuence the behaviour of the system. Here we restrict the analysis to how imperfections modify the string dynamics. In particular, we focus on two kinds of imperfections 7, ı.e., imperfections on the hopping parameter J and imperfections of the on-site energy (see discussion in Sec. 5.6). In Fig. 5.15 we report results for the case N = 4 and we compare

7Here we mean static imperfections. 5.7. Analysis of unitary errors 119

Jt = 0 Jt = 1

1.0 1.0

0.8 0.8

0.6 P 0.6 P 0.4 0.4

0.2 0.2

0.0 0.0

4 4

3 3

2 2 0 j 0 j 1 1 2 1 2 1 3 3 i 4 0 i 4 0

(A) Jt = 0. (B) Jt = 1. Jt = 3 Jt = 6

1.0 1.0

0.8 0.8

0.6 P 0.6 P 0.4 0.4

0.2 0.2

0.0 0.0

4 4

3 3

2 2 0 j 0 j 1 1 2 1 2 1 3 3 i 4 0 i 4 0

FIGURE 5.14: Screenshots of wavefront at different times. The bars represent the probability of finding a particle in the corresponding position. the ideal time evolution of the average position of the particle on the central track with the disordered one. We have taken J to vary by roughly 10%. In Fig. 5.15a we see that this amount of disorder slightly influences the dynamics of the string which still reaches the center with approximately the same velocity as the ideal case. We report that no-localization is observed also for larger (unrealistic) disorder of order σ/J = 10. The situation is different for the case of disorder on the on-site energy as shown in Fig. 5.15b. We see that increasing the ratio between the standard deviation and the hopping parameter σ/J we pass from a configuration in which there is essentially no-localization for σ/J = 0.1 to a localized configuration for σ/J = 10.0 8. This tells us that in order for the string to propagate we need an on-track variation of the on-site energy that is ≤ J. Finally, Fig. 5.16 shows the long-time behaviour of the position of the central particle for the ideal and the case with disorder on the on-site energy, which is taken to be at the "transition" σ/J = 1. In the ideal case, while on average the particle is in the middle, it still presents oscillations inherent to the unitary dynamics. The averaged disorder evolution instead presents little oscillations9 and we also notice that the average position is not exactly in the middle, but slightly closer to the initial position, showing a sort of memory effect.

8We report the same behaviour when we consider noise on the attractive interaction strength ∆. 9This is expected since we are taking a statistical average. 120 Chapter 5. Hamiltonian quantum computing

(A) Ideal time evolution vs. average time evolution with 10% gaussian disorder on the hopping parameter J.

(B) Ideal time evolution vs. time evolution with 10% gaussian disorder on the on-site energy.

FIGURE 5.15: Ideal vs. disordered time evolution. The disordered data are averaged over 50 simulation runs. The faded areas represent the standard deviation of the disordered curves. 5.7. Analysis of unitary errors 121

FIGURE 5.16: Ideal long-time time evolution vs. on-site disordered long time evolution.

J/ = 1/10 0.010

0.8 0.008

0.6 0.006 P S E

P 0.4 0.004

0.2 0.002

0.0 0.000 0 1 2 3 4 5 6 7 8 9 10 Jt

FIGURE 5.17: Example of probability of success (blue line, left axis) and error (red line, right axis) as a function of time for J/∆ = 1/10.

We ﬁnally point out that the issue of localization is also present in the context of general quantum walks (see Refs. [170, 171, 172]).

5.7.3 Errors in the CNOT logic

We now analyze the effect of unitary errors in our scheme, which originate from the fact that the correct logic is ensured only perturbatively. In particular, we consider a single CNOT, like the one in Fig. 5.4, and simulate it numerically. In our simulations we remove the spectator track on the bottom of Fig. 5.4. We then initialize the control and the target on the left sites, and in one of the internal basis states. Our main goal is to evaluate the probability that the CNOT is implemented succesfully or incorrectly, giving rise to an error in the logic. To this end we are interested in the probability that the particles are found on the right with the correct CNOT logic implemented on the internal state. We are interested in particular in relatively short times ∼ 1/J. We show a typical time evolution in Fig. 5.17, where we plot the probability of success PS and of error PE. In particular, the error probability is defined as the probability of finding both particles on the right side, but with incorrect internal states according to the CNOT logic, and we define analogously the success probability with the correct state instead. The probability of not finding the particles both at 122 Chapter 5. Hamiltonian quantum computing

Fit (J/ )4 10 4 PE

10 5 E P 10 6

10 7

10 2 10 1 J/

FIGURE 5.18: Time-averaged probability of error as a function of J/∆ for a CNOT after Jt = 3. The error scales with (J/∆)4. the end is the probability of an inconclusive measurement. We see that the probability of success quickly increases reaching a high maximum above at ≈ Jt = 3, while the error stays relatively low, although we start to see an increase at the end of the simulation, reaching ≈ 1%. We expect the error to decrease when we decrease the ratio J/∆. This is con- ﬁrmed in Fig. 5.18 in which we see that after a short time Jt = 3 the logical error caused by the CNOT scales as (J/∆)4. We point out that this scaling is insensitive to the choice of Jt It is important to understand that the CNOT construction works in the assumption that not only the perturbation theory is satisﬁed, but also that the time of the computation is not too long, or better a forward motion of the computation is guar- anteed. In fact, although the way the CNOT is implemented puts essentially a barrier between the correct and incorrect states, both states have the same energy, and we expect that after long times there is a high probability to tunnel to incorrect states. In- deed, it turns out that by averaging over very long times the probability of error and success is exactly the same. This is shown explicitly in Fig. 5.19 where for Jt ≈ 100 the time-averaged probability of success and error become comparable, and more and more equal increasing the time of observation. 5.7. Analysis of unitary errors 123

J/ = 1/10

0.35 PE PS 0.30

0.25

0.20 P 0.15

0.10

0.05

0.00 0 250 500 750 1000 1250 1500 1750 2000 Jt

FIGURE 5.19: Long time behaviour for the average probability of error and success.

125

Chapter 6

Towards an implementation with superconducting qubits

This chapter contains material by the author from Ref. [131] that is in preparation for publi- cation.

6.1 Dual rail encoding

We now show how the models presented in Chap.5 can be mapped to qubits using dual rail encoding. We have discussed the models in terms of particles with an internal spin degree of freedom. However, the particular commutation relations do not play a role in our problem, since the Hamiltonian does not involve exchange interactions between particles. This means that while the picture of a hopping particle with an internal degree of freedom is quite convenient for building intuition, we can also see the problem as if at each vertex of our lattices there is a qutrit (a three-level system). Each qutrit can be in the vacuum state |vaci, which represents the absence of the particle, or in the computational states |0i, |1i. Saying that there is only one excitation per track means that we are always considering superpositions of states in which only one qutrit per track is in one of the computational states, while all the others are in the vacuum state. Once we have this picture in mind it is quite easy to map a qutrit to two qubits using dual rail encoding. We denote by |gi and |ei the ground and excited states of each qubit. The state |ggiν represents the absence of the particle at site ν, the state |egiν represents a particle that is present at site ν in the computational state |0i, while |geiν in the computational state |1i. The state

|eeiν is not used. otice that we are implicitly assuming that each qubit has a certain frequency, which in the language of Sec. 5.6 corresponds to a certain on-site energy. Also from the analysis in Sec. 5.6,it follows that the two dual-rail encoded qubits need to have the same frequency. † In this mapping each creation operator as [ν] at a certain site ν with internal state + s = 0, 1 is substituted by a single-qubit σs [ν] operator, and analogously an annihi- − lation operator as[ν] by σs [ν]. Considering Eq. 5.25 we then realize that two-qubit 126 Chapter 6. Towards an implementation with superconducting qubits

ﬂip-ﬂop interactions of the form

0 + − hs | Up |si σs0 [i + 1, j + 1]σs [i, j] + h.c., (6.1) are needed to implement a matrix element of the single-qubit gate U in our model. In this representation, the coupling coefﬁcient of the interaction is the matrix element itself. It is then important that in an implementation that uses the dual rail mapping, we have the ability to control at least the sign of the coupling so that, for instance, we can implement a Hadamard gate. Moreover, the edge interactions are also readily obtained in the dual rail language. In fact a generic number operator at a site ν with internal state s = 0, 1 maps to 1

z ns[ν] → |ei he|s [ν] = 1/2(I + σs [ν]) (6.2)

. So a generic attractive interaction can be represented as

∆ H = −∆n[ν]n[ν0] → − (2I + σz[µ] + σz[µ])(2I + σz[ν] + σz[ν]). (6.3) e 4 0 1 0 1

Thus, at each edge we need to engineer four ZZ interactions with large negative co- efﬁcient −∆, involving the two physical qubits at each vertex. We point out however that the fact that we want the interactions to be negative is in this mapping purely conventional. If the sign were positive, the subspace of valid strings would now be the highest excited subspace of Hvalid and the perturbation theory would work in the same way. This equivalence is even clearer in the dual rail encoding since what we call the groundspace will always be for us the groundspace for the Hamiltonian reduced to the subspace with one excitation per track, while the true groundspace will always be the vacuum, in which there are no particles. This problem is in practice absent when we consider a real particle, like an electron. Due to dissipative processes the system will always and inevitably end up in the vacuum. Thus, the dual rail encoding introduces the problem of leakage out of the computational subspace, which limits the size of the quantum circuits that we can try to implement. In the next subsection, we try to understand in which way this problem scales with the size of the system.

6.1.1 Loss of particles

In the dual rail encoding every qubit would be subjected to decay processes (T1 processes, see Subsec. 3.3.4), which cause loss of excitations. We now show how this problem can be analyzed with a simple model. We actually formulate it in terms of a hopping particle that hops on a line with L sites. We associate to each site a decay rate that destroys the particle, ı.e., it brings the system from one of the sites |ki to a vacuum state |vaci. This means mathematically that we consider the following

1Here for concreteness we consider a site that is not split like in the CNOT or the Toffoli. 6.1. Dual rail encoding 127 master equation for the density matrix of our system (¯h = 1)

dρ N = −i[HL, ρ] + ∑ γD[|vaci hk|]ρ, (6.4) dt k=1

† † with the Lindblad dissipator D[A]ρ = A ρA − 1/2{A A, ρ} and HL the Hamilto- nian of a quantum walk on a line given by Eq. 5.55. We deﬁne the operator that counts whether there is an excitation or not

L ne = I − |vaci hvac| = ∑ |ki hk| . (6.5) k=1

We want to obtain the equation satisﬁed by the average of ne, ı.e., hnei(t) = Tr(ρ(t)ne), with the density matrix ρ(t) satisfying Eq. 6.4. We get

We notice that the operator ne commutes with the Hamiltonian HL, [ne, HL] = 0. This implies that the ﬁrst term on the right hand side of Eq. 6.6 is zero, since

Tr HL, ρ(t)]ne = Tr [ne, HL]ρ(t) = 0. (6.7)

Additionally, ne |vaci = 0, [ne, |ki hk|] = 0 and ne |ki hk| = |ki hk|. Using these relations we obtain L dhnei = − ∑ γTr |ki hk| ρ(t) = −γhnei, (6.8) dt k=1 which clearly shows that the average number of excitations on the line decays exponentially with decay rate γ. It should be noted that in order to derive rigorously this result it is necessary to assume that all the sites have the same decay rate, which in practice is only approximately true. However, this simple model allows us to understand that the characteristic decay rate of the excitation does not scale with the number of sites. This means in our lattice models that the problem of loss of particles per se scales with the number of tracks (excitations) and not with the number of applied gates. However, the number of applied gates governs in general the time of the simulation. From these reasoning we understand that in order to evaluate roughly the feasibility of a quantum circuit we can take as parameter ξ = NtrackγTsim, where

Tsim is the estimated time of the simulation, which depends on the parameters of the Hamiltonian and on the number of applied gates. The smaller the parameter ξ, the higher the probability that the system arrives succesfully at the end of the simulation. 128 Chapter 6. Towards an implementation with superconducting qubits

6.2 Implementation with superconducting qubits

In this section we describe a possible implementation of the schemes for Hamilto- nian quantum computing analyzed in Chap.5 in terms of transmon qubits. As we have shown in Sec. 6.1 the crucial point is to use the dual rail encoding. In practice, we need to engineer strong ZZ interactions, and ideally negative, as well as hopping interactions with at least a sign change. In transmon qubits the ZZ interactions correspond to cross-Kerr interactions, which can be conveniently expressed in terms of number operators, in a form that resembles the one we used in Chap.5. For this reason we will use this notation in what follows. Achieving the control over these kinds of interactions is also the general goal of analog quantum simulations of spin systems using artiﬁcial qubits [173]. The practical challenge is not only to get each of these interactions singularly, but to obtain all of them together, and most of all without unwanted spurious terms. The unwanted terms will in a realistic implementation always be there, so our goal will be to cope with their presence by making sure that they are smaller than the desired terms. We will consider an architecture in terms of grounded transmon qubits, in which each interaction is implemented by a single coupling element, either directly or indirectly. Additionally, since one of the advantages of HQC is the fact that no time-dependent control is needed in the bulk of the lattice, we forbid as a design principle the use of microwave drives to engineer the interactions. However, we will allow the presence of constant ﬂuxes.

6.2.1 Cross-Kerr interactions

The cross-Kerr interactions (ZZ) can be considered the most challenging to get mainly because we need to get them large compared to the hopping parameter. The size of the cross-Kerr interactions limits consequently the size of the hopping parameter, which eventually controls the speed of the computation in any of the models we discussed in Chap.5. As we said, the main problem in our implementation is caused by the loss of excitations, which is quantified by the typical T1 time of the qubits. We thus want the hopping parameter to be larger than the inverse of the T1 time to hope to be able to carry out the computation. There are several proposals for the implementation of cross-Kerr interactions in the literature [174, 175, 176, 177, 178, 179, 180]. We will build on ideas ideas similar to Refs. [176, 177, 178, 179]. The idea in these proposals is simply to use a Joseph- son junction in parallel with a capacitance. This element gives naturally rise also to unwanted hopping interactions. However, the capacitance and the junction give hopping of opposite signs and so it is possible to find a configuration in which the hopping is perfectly cancelled, which is basically the same working principle of an LC filter [48]. This scheme suffers from the problem of unwanted long-range coupling, which is a general feature of superconducting qubits architectures. This effect can be moderated by taking smaller coupling capacitances, so that the inverse of the 6.2. Implementation with superconducting qubits 129

EJ, Ca

Φext

αEJ, Cb Φ1 Φ2 L

CJ EJ1 CJ EJ2

FIGURE 6.1: Cross-Kerr coupler between two transmons on different adjacent tracks. The different color of the transmons denotes that they can have different frequencies. capacitance matrix remains approximately local, but this limits the achievable cross- Kerr interaction that can be achieved without unwanted hopping compared to the case in which we have two transmons alone. In the model we want to implement the problem is particularly serious for the following reasons. The unwanted terms will give a contribution to the hopping between transmons on the same line and thus give also a contribution to the desired hoppings that implement the matrix elements of a single-qubit gate, and so they should be taken into account in a full calculation. Even more importantly we expect also to get hopping couplings between qubits at the same site, which is definitely something we do not want. In order to limit these problems we consider an alternative coupler based on the following idea. We want a coupler that has inductive contribution smaller than a typical Josephson junction, but with the non-linear quartic term that remains approximately of the same order of magnitude. If this happens we expect the capacitance needed to filter the unwanted hopping to be much smaller than the one needed in the case in which we use a simple junction. We consider the circuit in Fig. 6.1. It consists of a Josephson junction in parallel with a simple inductance and an array of NJ junctions. Since this coupler provides the attractive interaction between two transmons located on different tracks the transmons can be taken to be off-resonant, as represented by the choice of different colors. This means that even if we have a hopping coupling between these transmons, it might still be ineffective. So the goal is not necessarily to cancel the hopping coupling perfectly, but to operate in a regime in which it is much smaller than the frequency detuning between the transmons. The array of equal Joseph- son junctions is assumed to be operated in the regime in which EJ/ECa 1 with 2 ECa = e /(2Ca) the charging energy of each junction. Additionally, we assume that the resonant frequencies of the array2 is much larger than any other frequency in the problem and we also assume capacitances to ground to be negligible. This is the same working regime of the superinductances used for the fluxonium qubits [61]. If these approximations are satisfied the internal degrees of freedom of the array can

2They are of the order of the plasma frequency of each junction. 130 Chapter 6. Towards an implementation with superconducting qubits be effectively eliminated and we can model the array as an effective potential

ϕ + 2πm Uarray,m(ϕ) = −NJ EJ cos , (6.9) NJ with ϕ the superconducting phase difference across the element and the parameter m ∈ {0, 1, . . . , NJ − 1} [62, 181, 182, 183]. From Eq. 6.9, we see that the effective Hamiltonian depends on the parameter m which labels the different metastable minima of the array of junctions in the limit EJ/ECa → +∞. In the limit EJ/ECa > 1, the array of junctions can be treated with the phase slip model, in which to each metastable minimum we associate a quantum state |mi with energy given by Eq. 6.9 [184]. The index m can be interpreted as the number of 2π turns that the phase of a

Josephson junction underwent (vortices). The ﬁnite EJ/ECa allows for coherent transitions from state |mi only to its nearest-neighbours |m ± 1i, in a ﬁrst approximation.

However, these transitions are exponentially suppressed with the ratio EJ/ECa 1. Thus if we work on this regime we can assume that the index m does not change. We will assume that the array is initially in the state |m = 0i so that the potential is given by (see also Ref. [185])

ϕ Uarray,m(ϕ) = −NJ EJ cos . (6.10) NJ

We should also take into account that the array adds an additional capacitance in parallel that is equal to the series capacitance of the array Ca/NJ. Thus the total coupling capacitance is given by Cc = Ca/NJ + Cb, with Cb the capacitance of the black sheep junction. It is worth mentioning that a system composed of an array of three large junctions in parallel with a black sheep junction has been analyzed in Refs. [186, 187, 188], and nicknamed the SNAIL. The goal there was to obtain a potential that gives rise to a three-wave mixing term (third-order), but without cross-Kerr (fourth-order), which is the unwanted term in this case. Here instead we would like to limit the quadratic term, while keeping the cross-Kerr relatively large. Additionally, in Ref. [185] a similar system was proposed for obtaining longitudinal qubit-resonator coupling. We now start the mathematical analysis of the circuit in Fig. 6.1. The Lagrangian of the circuit reads 3

C C C L = J Φ˙ 2 + J Φ˙ 2 + c (Φ˙ − Φ˙ )2 − U (Φ , Φ ), (6.11) 2 1 2 2 2 1 2 tot 1 2 with the total potential

2π 2π Utot(Φ1, Φ2) = −EJ1 cos Φ1 − EJ2 cos Φ2 + Uc(Φ1 − Φ2). (6.12) Φ0 Φ0

3We assumed the capacitances of the transmons to be equal, but this is not a necessary assumption. 6.2. Implementation with superconducting qubits 131 and the coupling potential

1 2 2π Uc(Φ1 − Φ2) = Φ1 − Φ2 − αEJ cos (Φ1 − Φ2) − 2L Φ0 2π Φ1 − Φ2 + Φext NJ EJ cos . (6.13) Φ0 NJ

The conjugate variables are obtained as Q1,2 = ∂L/∂Φ˙ 1,2. Taking the Legendre transform of the Lagrangian we obtain the Hamiltonian

2 2 Q1 Q2 Q1Q2 H = + + + Utot(Φ1, Φ2), (6.14) 2C˜ J 2C˜ J C˜c where we deﬁned the equivalent capacitances

1 C + Cc = J , (6.15a) ˜ 2 CJ CJ + 2CJCc

1 C = c . (6.15b) ˜ 2 Cc CJ + 2CJCc

Introducing the number of Cooper pairs Q1,2 = 2eN1,2 and the superconducting phases ϕ1,2 = 2πΦ1,2/Φ0, as in Subsec. 2.6.2 we rewrite our Hamiltonian as

2 2 H = 4EC N1 + 4EC N2 + 8EI N1N2 + Utot(ϕ1, ϕ2), (6.16) with charging energies 2 EC = e /(2C˜ J), (6.17a)

coup 2 Ecap = e /(2C˜c). (6.17b)

The problem is quantized by promoting Nm and ϕm to operators with commutation relation [ϕm, Nl] = iδml as explained in Chap.2. We now focus on the coupling potential given in Eq. 6.13, which in terms of superconducting phases reads EL,c 2 ϕ + ϕext Uc(ϕ) = ϕ − αEJ cos ϕ − NJ EJ cos , (6.18) 2 NJ where ϕ = ϕ1 − ϕ2, and we introduced the inductive energy

2 2 EL,c = Φ0/(4π L). (6.19)

In what follows it will be convenient to introduce a parameter β that is the ratio between inductive energy and Josephson energy of the array β = EL,c/EJ. Addi- 4 tionally, we ﬁx the external ﬂux to the value ϕext = 2πΦext/Φ0 = NJπ . We will now assume that it is possible to Taylor expand the coupling potential up to fourth order, as usually done for transmon qubits. This is a good approximation as long as

4In case of imperfections this parameter can be tuned. 132 Chapter 6. Towards an implementation with superconducting qubits

we work in the transmon regime EJ/EC 1 and we guarantee that the total potential has a deep global minimum at the point we are expanding, which is a condition that should always be checked. Expanding the coupling potential

1 1 1 1 ( ) = − + + + − 2 − − 4 + O( 6) Uc ϕ /EJ α NJ α β ϕ α 3 ϕ ϕ . (6.20) 2 NJ 24 NJ

We see that by setting the parameter γ = α + β − 1/NJ = 0 the quadratic term vanishes completely, while the fourth-order term would be approximately the same as the case with a simple junction, since the contribution of opposite sign given by 3 the array scales as 1/NJ . One should keep in mind that this point is a maximum of the coupling potential, but always (at least) a relative minimum of the total potential. This is due to the fact that the Hessian of the total potential is positive at this point, but the second derivative of the coupling potential is zero. As we will see, we would like the parameter γ not to be exactly zero, but sligthly larger in order to cancel perfectly the effective hopping when we project onto the qubit subspace. Hence, we will never work on this point. It is convenient to deﬁne the following energies coup 1 Eind = EJ α + β − , (6.21a) NJ

1 coup = − EKerr EJ α 3 . (6.21b) NJ We now perform a Taylor expansion for the total Hamiltonian Eq. 6.16 and rewrite it as 2 H = ∑ Hm + Hlin + HCK + Hnon.lin, (6.22) m=1 where we identiﬁed the Hamiltonian of the single transmons

E E H = 4E N + Lm ϕ2 − Km ϕ4 , (6.23) m C m 2 m 24 m with coup ELm = EJm + Eind , (6.24a) coup EKm = EJm + EKerr , (6.24b) with m = 1, 2. The linear coupling Hamiltonian Hl is given by

coup coup Hl = 8Ecap N1N2 − Eind ϕ1 ϕ2, (6.25) while the term responsible for the cross-Kerr is

Ecoup H = − Kerr ϕ2 ϕ2. (6.26) CK 4 1 2 6.2. Implementation with superconducting qubits 133

We also introduced what we called a non-linear Hamiltonian

Ecoup H = Kerr (ϕ3 ϕ + ϕ ϕ3). (6.27) non.lin 6 1 2 1 2

We introduce annihilation and creation operators for the transmon modes again in analogy with Subsec. 2.6.2

1/4 2EC † ϕm = am + am , (6.28a) ELm

1/4 i ELm † Nm = am − am . (6.28b) 2 2EC In second quantized form and performing several rotating wave approximations (RWA) the previous Hamiltonians read:

• Transmon Hamiltonian

RWA δ H = h¯ (ω + δ )a† a + h¯ m a† a† a a , (6.29) m m m m m 2 m m m m √ with the transmon frequency ωm = 8ECELm/¯h and the anharmonicity δm =

−EKmEC/(hE¯ Lm).

• Linear coupling Hamiltonian gives

RWA † † Hlin = hJ¯ lin(a1a2 + a1a2), (6.30)

with the linear hopping parameter Jlin given by

Jlin = Jcap + Jind, (6.31)

where we identiﬁed a capacitive Jcap and an inductive Jind contribution

1/4 1/4 1 coup EL1 EL2 Jcap = 2Ecap , (6.32a) h¯ 2EC 2EC

1/4 1/4 1 coup 2EC 2EC Jind = − Eind . (6.32b) h¯ EL1 EL2 In these equations we clearly see that capacitive and inductive couplings give a hopping of different sign.

• Cross-Kerr Hamiltonian RWA 1 1 1 1 H = hJ¯ a†a a†a + a†a + a†a + a a a†a† + a†a†a a , (6.33) CK CK 1 1 2 2 2 1 1 2 2 2 4 1 1 2 2 4 1 1 2 2 134 Chapter 6. Towards an implementation with superconducting qubits

with 1/2 1/2 1 coup 2EC 2EC JCK = − EKerr = −∆. (6.34) h¯ EL1 EL2

• Non-linear Hamiltonian

RWA † † † † † Hnon.lin = hJ¯ non.lin,1[(a1a2 + a1a2) + (a1a2 + a1a2)a1a1+ † † † † † † † a1a1(a1a2 + a1a2) + (a1a2a1a1 + a1a1a1a2)]+ † † † † † hJ¯ non.lin,2[(a1a2 + a1a2) + (a1a2 + a1a2)a2a2+ † † † † † † † a2a2(a1a2 + a1a2) + (a1a2a2a2 + a2a2a1a2)], (6.35)

with coup 3/4 1/4 1 EKerr 2EC 2EC Jnon.lin,1 = (6.36a) h¯ 6 EL1 EL2 coup 1/4 3/4 1 EKerr 2EC 2EC Jnon.lin,2 = (6.36b) h¯ 6 EL1 EL2

H Q = Ω n + Ω n + J (σ+σ− + σ−σ+) − ∆n n , (6.37) h¯ 1 1 2 2 hop 1 2 1 2 1 2 with Ωm = ωm + δm − ∆/2, while the total hopping is

Jhop = Jcap + Jind + Jnon.lin, (6.38) with Jnon.lin = 3(Jnon.lin,1 + Jnon.lin,2). In Eq. 6.37 the number operators have to be taken as in Eq. 6.2.

6.2.2 Hopping interaction

As we know in our scheme hopping interactions need to be engineered between qubits on the same track, which are resonant or approximately resonant. In Subsec. 6.2.1, we have seen that the hopping interaction is naturally obtained with transmon qubits, by simply using capacitances or inductances. The order of magnitude of the hopping interactions need to be much weaker than the cross-Kerr interactions on the edges. Considering capacitive coupling between equal transmons as shown in Fig. 6.2, the general functional form of the coupling parameter can be deduced from the analysis done in Subsec. 6.2.1 and it is given by (see Eq. 6.31)

1/2 1 coup EL Jcap = 2Ecap , (6.39) h¯ 2EC

5We expect the presence of higher levels to give a small correction to the effective hopping coupling, since we are already assuming the direct hopping coupling to be weak. 6.2. Implementation with superconducting qubits 135

FIGURE 6.2: Capacitive coupling between two transmons on the same track. The fact that the transmons have same color denotes that they have the same frequency.

where the effective EL and EC will always be dependent on the site we are choosing. We see that capacitive coupling always gives coupling of the same sign. This is not sufﬁcient if we want to achieve universality via Toffoli and Hadamard gate [163, 164], and in general with any set of universal real gates, since we need at least a sign change in one of the fundamental gates. An immediate solution would be to use inductances to implement couplings of negative signs. However, in order to keep the coupling small, one needs to go immediately in the regime of large inductances, and eventually use an array of junctions. There is a simpler way to achieve this sign change, ı.e., to couple both qubits capacitively to a common resonator, and assume the qubits to be far detuned from the resonators’ frequencies, working in the so- called dispersive regime, that we have already encountered in Chap.3 and Chap.

4. By calling gr the coupling to the resonator of the two qubits, and assuming it for simplicity to be equal for both qubits, δ their anharmonicity and Ω their frequency we obtain the effective hopping coupling between the two qubits [13, 51] √ 2 2 gr 1 ( 2gr) Jeff = − , (6.40) Ω − ωr 2 Ω + δ − ωr with ωr the frequency of the resonator. It is then clear that we can obtain coupling of different signs by properly selecting the frequency of the resonator. For instance by taking ωr > Ω and δ < 0, we would get a negative effective hopping as expected.

bare EJ1/EC 80 bare EJ2/EC 60 bare ECa/EC 12 bare EJ/EC 1200 bare EL/EC 255

bare TABLE 6.1: Parameters in units of EC . 136 Chapter 6. Towards an implementation with superconducting qubits

5000

4950

4900 e r 4850 a b C E /

t 4800 o t U 4750

4700 3 2 4650 1 0

1 3 1 2 1 2 0 1 2 3 2 3

FIGURE 6.3: Total potential Utot(Φ1, Φ2) for the parameters in Table 6.1. α is set to 0.042.

It is worth commenting on the fact that assuming the hopping to be obtained from passive capacitive coupling, ı.e., by not considering the possibility of external drives, and assuming to work in the transmon regime we are giving up on the possibility to have complex coupling in the basis we have chosen. This is a consequence of the fact that the Hamiltonians we used to obtain the couplings are invariant under a time- reversal transformation, and additionally, the basis we have chosen is the eigenbasis of an operator invariant under time-reversal symmetry, that is the Hamiltonian without coupling. In more detail the capacitive coupling Q1Q2 and also the inductive coupling Φ1Φ2 are invariant under the transformations Qm → Qm, Φm → −Φm [189]. It is quite interesting to note that in our model this observation is directly connected to the fact that we cannot achieve the standard universality with CNOT, H and π/8 gate, but we can achieve universality with Toffoli and H. In principle, not only Hadamard and Toffoli, but also all single-qubit real gates can be obtained, as well as controlled-U and controlled-controlled-U with U real. We ﬁnally point out that passivity, meaning no external time-dependent drive, does not imply time-reversal invariance in circuit QED systems as shown in Refs [190, 191].

6.2.3 Estimates of parameters and trade-offs

There are several trade-offs that one should consider in our implementation. First of all, we know that we need the ratio EJ/ECa of each junction of the array to be much larger than one, in order to prevent phase slips. We will always ﬁx EJ/ECa = 100, which is a typical standard value [185]. However, we know that we want a small 6.2. Implementation with superconducting qubits 137

0.5

0.0

0.5

1.0 bare Jcap/EC bare 1.5 Jind/EC bare Jnon. lin/EC 2.0 bare Jhop/EC

0.042 0.044 0.046 0.048 0.050

(A) Strength of the different contributions to the hopping parameter (dashed lines) and total hopping parameter (solid line) as deﬁned in Eq. 6.38. 0.7

0.8

0.9 /Ebare 1.0 C bare 1/EC 1.1 bare 2/EC 1.2

1.3

1.4 0.042 0.044 0.046 0.048 0.050

(B) Cross-Kerr coupling and anhrmonicities of the two transmons.

FIGURE 6.4: Hopping (a) and cross-Kerr (b) coupling parameters for α close to the condition γ = 0. We see that the hopping coupling is zero at α ≈ 0.043, corresponding to a Josephson energy of the small bare junction equal to αEJ = 51.72 in units of EC . 138 Chapter 6. Towards an implementation with superconducting qubits

FIGURE 6.5: Two equal transmons (red) coupled to a common transmon (green) via the coupler analyzed in Subsec. 6.2.1. Parameters are not shown for simplicity. effective coupling capacitance in order to avoid large unwanted coupling between next-nearest neighbour qubits. To reduce this capacitance we could also increase the number of junctions, which we would like to limit from a pratical point of view. In what follows, we just assume a set of typical and reasonable experimentally achievable parameters, and compute the typical hopping and also the nearest-neighbour induced hopping. In particular, to evaluate the order of magnitude of the nearest- neighbour hopping, we just consider two identical transmons coupled via our coupler to a common transmon as in Fig. 6.1, and compute the hopping between the two identical transmons, by repeating the previous quantization procedure. We will focus on the case NJ = 4, which however should not be taken as an optimized value, bare 2 and take the bare charging energy of the transmons EC = e /(2CJ) as our unit of energy and also seth ¯ = 1. We neglect the intrinsic capacitance of the coupling junction Cb = 0 so that all the coupling capacitance is given by the equivalent capacitance of the array of junctions. The parameter we chose are given in Table 6.1. In Fig. 6.3 we plot an example of the total potential showing that it has a global minimum in the origin, which holds true for all the parameters we have chosen. In Fig. 6.4a we see how it is possible to achieve a hopping term that is exactly equal to zero. This is possible with an effective capacitance that is now much smaller than the transmon capacitance, in particular

Cc ≈ 1/(48)CJ. We can estimate the next-nearest neighbour interaction of two resonant transmons coupled to a common transmon, via the cross-Kerr coupler analyzed in Subsec. 6.2.1, by considering the circuit in Fig. 6.5. Repeating the analysis presented in Subsec. 6.2.1, and using the previous reasonable parameters, the estimated bare 6 next-nearest-neighbour hopping is ≈ 0.004EC . Considering that we can get quite bare high ∆ ≈ 0.8EC as we can see from Fig. 6.4b, we can choose the desired hop- bare ping parameter to be for instance Jhop ≈ ∆/10 ≈ 0.08EC and take into account the smaller contribution of the ideally unwanted next-nearest neighbour coupling in the calculation. 6 We point out that these are estimates that give only the order of magnitude of the next-nearest neighbour hopping when we consider the coupler in the whole lattice. 6.3. Full implementation 139

6.3 Full implementation

Flip-ﬂop coupling J 2π × 10 MHz 1 2 1 2 (Jσ+σ− + σ−σ+) Transmon cross-Kerr coupling ∆ 2π × 100 MHz (−∆a†ab†b)

Transmon frequency detuning between ≈ 500 MHz adjacent tracks

Total transmon freq. bandwidth ≈ 1.5 GHz

TABLE 6.2: Estimates of typical parameters achievable in our implementation.

In the previous part of this chapter we have discussed how the models introduced in Chap.5 can be mapped to qubits via dual rail encoding and how the basic interactions can be engineered using speciﬁc coupling elements. In this section, we show qualitatively how these elements have to be put together in order to realize the full models. We will discuss this for the Lloyd-Terhal lattice initially. The layout of a small size implementation is depicted in Fig. 6.6, considering only single-qubit gates. At each site of the lattice we have two transmons that in dual rail encoding implement effectively a qutrit. We see that sites on different tracks can have different colors, which means that the on-site energy of sites of different tracks can be different as explained in Subsec. 5.6, while sites on the same track must have the same color, because the particles should not distinguish on which site they are. As discussed in Subsec. 5.7.2 we can tolerate variation of the on-site qubit frequencies on a track of the order of the hopping parameter. At any rate we can choose adjacent tracks to be far detuned from each other, which helps in mitigating the problem of unwanted hoppings between the lines. In Fig. 6.6 we have chosen a conﬁguration with three colors, which can then be repeated. As discussed in Sec. 6.2.2, in our scheme gates are applied by elements involving capacitances and resonators. In Fig. 6.7 we see an example of X and Hadamard H gate. The resonator in the Hadamard gate mediates the interaction between the qubits providing the minus sign needed for the diagonal matrix element of the Hadamard. The blue boxes in Fig. 6.6 represent the couplers needed at the edges in order to realize the strong attractive interactions of the model. Each blue box corresponds to four coupling elements as that in Fig.6.1, which couple the two transmons on a site with the two transmons on a site connected by an edge. As shown in Fig. 6.6 the system does not need any active control in the bulk. Active control is required only for initialization purposes and readout. The feedlines on the left of Fig. 6.6 put the excitations (particles) on the left vertices in a certain internal state, which can always be taken to be the state in which all dual rail qubits are in the computational |0i state. This means that at a certain site only one of the two transmons is in the excited state. The system evolves under the action of our engineered 140 Chapter 6. Towards an implementation with superconducting qubits

U U

U U U

U U

2 transmons zz coupler feedline

U circuit for gate U readout resonator

FIGURE 6.6: Layout concept for the Hamiltonian quantum computing scheme analyzed in this chapter with superconducting qubits.

X = (A) X gate circuit.

H =

(B) Hadamard gate circuit.

FIGURE 6.7: Examples of circuits for implementing single-qubit gates.

FIGURE 6.8: Circuit for realizing the strong ZZ attractive interactions. 6.4. Main challenges and practical considerations 141

U U U

FIGURE 6.9: Layout concept for the Feynman lattice.

Hamiltonian that implements the desired quantum circuit. After a certain time the qubits on the right side of the lattice (or in a measurement region) are measured via standard dispersive readout and the result of the computation extracted. Finally, the analogous implementation of the Feynman lattice discussed in Sec. 5.5 is shown in Fig. 6.9. In Table 6.2 we summarize the estimated relevant parameters.

6.4 Main challenges and practical considerations

The realization of a lattice of superconducting qubits like the one depicted in Fig. 6.6 places several experimental challenges. The coupler for obtaining the attractive interactions analyzed in Subsec. 6.2.1 requires the fine tuning of the parameters and of the flux biases in order to work in the desired regime. In addition, our architecture requires that the flux biases can be controlled on the desired superconducting loops without affecting other loops. This problem might be mitigated by ungrounding the transmons, ı.e., by inserting a capacitance to ground for each transmon. However, this automatically limits the achievable magnitude of the attractive interactions as it also happens in related couplers [192, 179]. While the ideal model requires all qubits on a track to have the same frequency (on-site energy), as shown in Subsec. 5.7.2 disorder on the qubits’ frequency can be tolerated from the point of view of the propagation of the computation if its standard deviation stays below the value of the typical hopping parameter 7. From this analysis we conclude that in our transmon implementation we can tolerate no more than 15 MHz disorder on the qubit frequencies on a track, which can be achieved by careful design.

7Here we are referring to the Lloyd-Terhal scheme.

143

Chapter 7

Conclusions

In this thesis we have explored the capabilities of superconducting qubits as a plat- form for quantum computation. In the first part of the thesis (Chap.2) we have presented an introduction to the basic tools for the analysis of superconducting circuits emphasizing the double role of superconductivity. Superconductors guarantee low dissipation, which is necessary for the observation of coherent quantum effects. In addition, the existence of a non-linear superconducting element, the Josephson junction, allows the creation of non-trivial quantum states, whose main example is a qubit. After the introduction to the main families of superconducting qubits, we moved to the analysis of systems based on transmons or transmon-like qubits. In Chap.3 we have analyzed in detail a scheme for performing direct three-qubit parity (or stabilizer) measurements, via multi-qubit dispersive readout originally conceived in Ref. [17]. We explained the importance of these kinds of measurement in the context of quantum error correction and also for generation of entanglement via measurement. In contrast to ancilla-based stabilizer measurements, direct parity measurements have the advantage that they do not require any intermediate two- qubit gates. However, in order to be considered true stabilizer measurements, direct parity measurements are required to reveal information only about the parity and not about the particular state of the qubits. This is a pure quantum requirement, which makes their design highly non-trivial. The direct scheme we analyzed is based on an ideal generalized dispersive Jaynes-Cummings Hamiltonian with three qubits and two resonators. The measurement consists in sending a microwave tone in the two cavities at a specific frequency, and by subsequent readout of the parity- dependent phase acquired by the signal. By means of input-output theory, we identified the condition that the system needs to satisfy in order to implement the measurement succesfully. In particular, we analyzed the important role that unwanted qubit-state-dependent coupling of the two resonators (quantum switch) plays in the measurement. In Chap.4, we showed that for transmon qubits capacitively coupled to the resonators, the quantum switch terms do not allow the parity measurement to be performed. However, we have shown that with tunable coupling qubits (TCQ), in which two capacitively coupled transmons form one qubit, we can achieve sufficient flexibility in the parameters so that not only, we can satisfy, what we called, the 144 Chapter 7. Conclusions parity measurement condition, but even cancel completely the coupling between the two resonators, realizing the ideal model. This is achieved in a quite singular situation in which the qubits are dark for one resonator, but not for the other. This configuration can be obtained with capacitive coupling, and it requires the ability to couple the transmons composing the TCQ at different locations of the resonators, considering a two-dimensional implementation. Although this condition represents the ideal case, we showed that the parity condition can be satisfied also in the case in which each transmon of the TCQ is coupled to only one of the two resonators. In the last part of the thesis we analyzed an alternative way of performing a quantum computation using a time-independent Hamiltonian. This idea is closely related to an original proposal due to Feynman in Ref. [18], and it was developed by Lloyd and Terhal in Ref. [19]. The scheme is initially formulated in terms of particles hopping on their own track, with attractive interactions that coordinate their motion, and reviewed in Chap.5. The quantum information is encoded in the internal spin degree of freedom of the particles, and quantum gates are associated with each hopping term in the Hamiltonian. In particular, we extended the analysis of Ref. [19] showing explicitly how to construct controlled-controlled unitaries such as the Toffoli gate. In addition, we also constructed a new model which, within the due approximations, maps unitarily to the one proposed by Feynman in Ref. [18], although it maintines the interactions relatively local. In this new model, for small scale circuits, ideas from perfect quantum state transfer can be employed to guarantee that the computation is completed with unit probability at a certain time. However, the original model of Ref. [19] has in general a more efficient dynamics, since its propagation is due essentially to entropic considerations. In Chap.5, we also performed some numerical analysis of the effect of imperfections in the model. In Chap.6 we proposed an implementation with superconducting transmon qubits. The mapping to qubits can be achieved via dual rail encoding. In this mapping the attractive interactions are implemented by strong cross-Kerr (ZZ) interactions between the transmons, while the hopping, and thus the quantum gates, by flip-flop interactions. We describe in details how it can be possible to implement these interactions with dedicated elements. In particular, we proposed a new coupling element for realizing the strong cross-Kerr interactions based on array of junctions. The coupler, being direct, allows to achieve large values of the cross-Kerr parameters, but it is designed to limit the cross-talk between qubits. In our implementation, the hopping can be simply realized with capacitive coupling, or via capacitive coupling mediated by a resonator when we need a sign change. In addition, in the transmon implementation only real gates are available, which makes the construction of the direct Toffoli described in Chap.5 quite relevant, since universality can be achieved with Toffoli and Hadamard gates. In Chap.6 we also provided some estimates of the typical parameters that might be achievable with current technology. 7.1. Outlook 145

7.1 Outlook

I am glad to say that there are several open and stimulating questions that this thesis bring up. I have an endless list of them, which this section is too narrow to contain.I will then just try to mention the main ones. The direct parity measurement we studied was limited to the three-qubit case, while quantum error correcting codes, such as the surface code, require also four- qubit parity measurements. In this case, the measurement would require three resonators and the effect of the quantum switch terms in this case has not been studied yet. These terms are also ubiquitous in any circuit QED implementation and are usually neglected, although they might give an important contribution. The author also believes that the configuration in which the TCQ cancels completely the quantum switch term, as discussed in Chap.4, gives rise to interesting physics, whose potential has not yet been studied. There are of course many specific spin-off works that can be envisioned from the Hamiltonian quantum computing scheme presented in Chap.5 and Chap.6. First of all, it is unclear how quantum error correction can be performed in this scheme, since QEC is usually formulated for the circuit model, and it is not straightorwardly generalizable to other models. The same problem is faced by adiabatic quantum computation. In addition, while in this thesis we analyzed how imperfections modify the dynamics of the computation, it is still an open question how they influence the implementation of a single gate. This is a problem that can be formulated also in terms of the original Feynman Hamiltonian, in which case we should consider that the matrix we are implementing is not unitary. The author found that this problem has not been fully addressed in the literature. In the implementation part more studies can be envisioned on the cross-Kerr coupler described in Chap.6, in particular exploring how the flux in the loop can be tuned to work at an optimal point in case imperfections are present in the system. Also, the problem of cross-talk in the complete architecture needs to be better understood. To this end the author thinks that ideas from Ref. [193] can be suited for a more complete analysis. Moreover, it is still unclear how the model can be made programmable, which seems to require the use of tunable couplers. As a speculation, it might be interesting to study how to conceive a quantum computation in which gates are implemented via single-elements like the ones described in Chap.5, which then have an interface with other standard qubits.

147

Appendix A

Transmission lines

In this Appendix we review the treatment of inﬁnite transmission lines and transmission line resonators in Lagrangian formalism. We point the reader to Refs. [16, 48, 194] for more detailed treatments and applications.

A.1 Inﬁnite transmission line

We start by considering the discrete model of an inﬁnite, lossless transmission line depicted in Fig. A.1. Using the procedure for obtaining the Lagrangian of a general circuit described in Chap.2 we obtain the following Lagrangian

+∞ c∆x +∞ 1 L = Φ˙ (k∆x, t)2 − [Φ((k + 1)∆x, t) − Φ(k∆x, t)]2. (A.1) ∑ ∑ ` k=−∞ 2 k=−∞ 2 ∆x

In the limit of ∆x → 0 the collection of fluxes ~Φ = {..., Φ(−∆x), Φ(0), Φ(∆x),... } effectively as a field,ı.e., ~Φ → Φ(x, t). The Lagrangian then becomes the Lagrangian of a massless scalar Klein-Gordon field

Z +∞ c ∂Φ 2 1 ∂Φ 2 Z +∞ L = dx − = dxL(∂tΦ, ∂xΦ). (A.2) −∞ 2 ∂t 2` ∂x −∞

The Hamiltonian is also readily obtained by deﬁning the conjugate ﬁeld

∂L ∂Φ q(x, t) = = c , (A.3) ∂(∂tΦ) ∂t

I Φ(−∆x) Φ(0) Φ(∆x) Φ(2∆x) Φ(n∆x)

`∆x

V c∆x

Φg = 0 ∆x

FIGURE A.1: Discrete model of a inﬁnite transmission line 148 Appendix A. Transmission lines which is the charge per unit of length, to get the Hamiltonian

∂Φ Z +∞ q(x, t)2 1 ∂Φ 2 H = q(x, t) − L = dx + . (A.4) ∂t −∞ 2c 2` ∂x

The Euler-Lagrange equation associated with the Lagrangian Eq. A.2 reads

∂ ∂L ∂ ∂L + = 0 (A.5) ∂t ∂(∂tΦ) ∂x ∂(∂xΦ) which gives the wave equation for the ﬁeld Φ(x, t)

∂2Φ ∂2Φ − v2 = 0, (A.6) ∂t2 p ∂x2 p with phase velocity vp = 1/(`c). The wave equation Eq. A.6 has solutions of the ← ← → → form Φ (x, t) = Φ (t + x/vp) and Φ (x, t) = Φ (t − x/vp). From the linearity of the wave equation the general solution is

→ ← → ← Φ(x, t) = Φ (x, t) + Φ (x, t) = Φ (t − x/vp) + Φ (t + x/vp). (A.7)

The wave equation is also satisﬁed by the voltage V(x, t) and current I(x, t) (and also by q(x, t) = cV(x, t)). This follows immediately from the relations

∂Φ 1 ∂Φ V(x, t) = , I(x, t) = − , (A.8) ∂t ` ∂x and from the assumptions that the partial derivatives commute. Therefore also V(x, t) and I(x, t) have solutions of the form Eq. A.7. We also deduce the so-called telegrapher’s equations ∂V ∂I = −` , (A.9a) ∂x ∂t ∂I ∂V = −c , (A.9b) ∂x ∂t which are just the continuous version of Kirchhoff’s laws for the transmission line. From the telegrapher’s equation we also obtain a relation between the left and right propagating voltages and currents, namely

V← V→ I← = − , I→ = . (A.10) Z0 Z0

The transmitted power (positive to the right in our convention) can be expressed as

(V→)2 (V←)2 P(x, t) = V(x, t)I(x, t) = − . (A.11) Z0 Z0

Our goal now is to introduce a mode expansion of the field Φ(x, t) and accordingly also of the voltage, current and charge fields. To this end we write Φ(x, t) in terms A.1. Infinite transmission line 149

of its spatial Fourier transform Φk(t)

Z +∞ 1 ikx Φ(x, t) = √ dke Φk(t). (A.12) 2π −∞

Since Φ(x, t) is real we immediately obtain the constraint

∗ Φk(t) = Φ−k(t). (A.13)

Notice that Φk(t) is in general complex and so when we pass to a quantum mechanical description we cannot associate a hermitian operator to it, but only to its real and imaginary part. Taking the Fourier transform of the wave equation Eq. A.6 we get the equation for Φk(t) ∂2Φ k + v2 k2Φ = 0, (A.14) ∂t2 p k which have to be satisﬁed ∀k. The general solution of Eq. A.14 is

−iωkt ∗ iωkt Φk(t) = Cke + Dk e , (A.15) with frequency ωk = vp|k| and Ck, Dk two constants that depend on the parameter k and the initial conditions. From Eq. A.13 it follows that we can just consider the problem for k ≥ 0. In particular, we can write

−iωkt ∗ iωkt Φk(t) = Cke + C−ke , (A.16)

∗ which we rewrite in terms of dimensionless amplitudes αk, α−k as s h ¯ −iωkt ∗ iωkt Φk(t) = αke + α−ke . (A.17) 2cωk

Plugging the previous equation into Eq. A.12 we write Φ(x, t) as in Eq. A.7 and we identify r h Z +∞ → ¯ 1 −iωk(t−x/vp) Φ (x, t) = dk √ αke + c.c. , (A.18a) 4πc 0 ωk r h Z 0 ← ¯ 1 −iωk(t+x/vp) Φ (x, t) = dk √ αke + c.c. . (A.18b) 4πc −∞ ωk

Now making a change of variable in the integrals ω = vp|k| we write Eqs. A.18 as r hZ¯ Z +∞ 1 Φ→(x, t) = 0 dω √ α→(ω)e−iω(t−x/vp) + c.c. , (A.19a) 4π 0 ω r hZ¯ Z +∞ 1 Φ←(x, t) = 0 dω √ α←(ω)e−iω(t+x/vp) + c.c. , (A.19b) 4π 0 ω 150 Appendix A. Transmission lines in which we deﬁned the new amplitudes

αω/v α→(ω) = √ p , (A.20a) vp

α−ω/v α←(ω) = √ p . (A.20b) vp We can now quantize the theory by promoting the complex amplitudes α→(ω) and α←(ω) to operators aˆ→(ω) and aˆ←(ω), satisfying commutation relations

aˆ→(ω), aˆ†→(ω0) = δ(ω − ω0) , aˆ←(ω), aˆ†←(ω0) = δ(ω − ω0). (A.21)

Assuming these commutation relations one can show that we recover correctly the standard ﬁeld quantization condition

Φˆ (x), qˆ(x0) = ih¯ δ(x − x0), (A.22) with r hZ¯ Z +∞ 1 Φˆ →(x) = 0 dω √ aˆ→(ω)eiωx/vp + h.c. , (A.23a) 4π 0 ω r hZ¯ Z +∞ 1 Φˆ ←(x) = 0 dω √ aˆ←(ω)e−iωx/vp + h.c. , (A.23b) 4π 0 ω and r hZ¯ Z +∞ √ qˆ→(x) = c 0 dω ω iaˆ†→(ω)e−iωx/vp + h.c. , (A.24a) 4π 0 r hZ¯ Z +∞ √ qˆ←(x) = c 0 dω ω iaˆ†←(ω)e+iωx/vp + h.c. , (A.24b) 4π 0 whose form is deducible from their classical expressions. In analogy with Eqs. A.20 † we also introduce the annihilation and creation operators aˆk, aˆk , which satisfy commutation relations † 0 aˆk, aˆk = δ(k − k ). (A.25)

Plugging Eqs. A.23 and A.24 into the Hamiltonian of the inﬁnite transmission line Eq. A.4 (in operator form) we obtain the Hamiltonian in diagonal form

Z +∞ Z +∞ ˆ †← ← †→ → † H = dω h¯ ω aˆ (ω)aˆ (ω) + aˆ (ω)aˆ (ω) = dk h¯ ωkaˆk aˆk. (A.26) 0 −∞

A.2 Finite transmission line

In this section we consider a transmission line of ﬁnite length that extends from

x = 0 to x = L0. Similarly to the previous section we obtain the Lagrangian as

2 2 Z L0 c ∂Φ 1 ∂Φ L = dx − (A.27) 0 2 ∂t 2` ∂x A.2. Finite transmission line 151 which again generates the Euler-Lagrange equation Eq. A.6. We solve this equation assuming open circuit boundary conditions

∂Φ ∂Φ I(0, t) = I(L , t) = 0 =⇒ (0, t) = (L , t) = 0. (A.28) 0 ∂x ∂x 0

Looking for non-trivial solutions (non-zero) of the form Φ(x, t) = ξ(t) f (x), inserting into Eq. A.6 and using the boundary conditions, we obtain a family of solutions parametrized by an integer

Φn(x, t) = ξn(t) fn(x), (A.29) n = {0, 1, 2, . . . }, with

fn(x) = An cos(knx). (A.30)

The choice of the constants An is actually arbitrary. However, we take them such 2 that the ` norm of the fn(x) is equal to one. This is deﬁned as

1/2 Z L0 1 2 k fnk`2 = dx fn(x) . (A.31) L0 0

Since for n ≥ 1 1 Z L0 πn 1 dx cos2 = , (A.32) L0 0 L0 2 √ we take An = 2 ∀n ≥ 1, while for n = 0, we take trivially A0 = 1. Compactly

f0(x) = 1, (A.33a)

√ πn fn(x) = 2 cos x . (A.33b) L0 From the linearity of the wave equation we can write a general solution of Eq. A.6, with boundary conditions given by Eq. A.28

+∞ Φ(x, t) = ∑ ξn(t) fn(x). (A.34) n=0

The variables ξn(t) satisfy now

d2ξ n = −ω2ξ (t), (A.35) dt2 n n which is the equation of motion of a harmonic oscillator with characteristic frequency ωn = vpkn = vpπn/L0. Plugging Eq.A.34 into the Lagrangian Eq.A.27 and using the properties 1 Z L0 dx fn(x) fm(x) = δnm (A.36a) L0 0

Z L0 1 d fn d fm 2 dx = knδnm. (A.36b) L0 0 dx dx 152 Appendix A. Transmission lines we obtain +∞ 1 ˙2 2 2 1 ˙2 L = L0c ∑ (ξn − ωnξn) + L0cξ0. (A.37) 2 n=1 2

The mode ξ0 is a free particle corresponding to a constant voltage (charge) and thus a linearly increasing flux. In an equivalent circuit, it just corresponds to a capacitance in series with all the other LC oscillators and can thus be taken into account exactly. We will neglect it from now on meaning that our flux is always the difference between Φ(x, t) and the linearly increasing flux. Neglecting the free particle mode, we obtain the conjugate variables

∂L qn = = L0c ξ˙n, (A.38) ∂ξ˙n and the Hamiltonian +∞ 2 qn L0c 2 2 H = ∑ + ωnξn. (A.39) n=1 2L0c 2 We now quantize the problem as usual by promoting variables to operators denoted by hats +∞ 2 ˆ qˆn L0c 2 ˆ2 H = ∑ + ωnξn, (A.40) n=1 2L0c 2 and imposing commutation relations between conjugate variables

[ξˆn, qˆn] = ih¯ . (A.41)

We notice that we just have the Hamiltonian of a collection of independent harmonic oscillators with equal "mass" L0c (the total capacitance) and frequencies ωn. We then introduce annihilation and creation operators s ˆ h¯ † ξn = (aˆn + aˆn) (A.42a) 2L0cωn

r h¯ qˆ = i L cω (aˆ† − aˆ ), (A.42b) n 2 0 n n n † which satisfy the commutation relations [aˆn, aˆm] = δnm. The Hamiltonian becomes

+∞ +∞ ˆ † h¯ ωn H = ∑ h¯ ωnaˆnaˆn + ∑ , (A.43) n=1 n=1 2 where the second term, although manifestly inﬁnite, is just a constant, which does not enter the equations of motion. In the Heisenberg picture s +∞ h¯ ˆ −iωnt † iωnt Φ(x, t) = ∑ cos(knx) aˆne + aˆne , (A.44) n=1 L0cωn A.2. Finite transmission line 153 from which we deduce the expression for the voltage operator in the Heisenberg picture s ∂Φˆ +∞ h¯ ω ˆ n † iωnt −iωnt V(x, t) = = ∑ i cos(knx) aˆne − aˆne , (A.45) ∂t n=1 L0c which in the Schrödinger picture is s +∞ ˆ h¯ ωn † V(x) = ∑ i cos(knx) aˆn − aˆn . (A.46) n=1 L0c

155

Appendix B

Universality of Hadamard, controlled-Hadamard and CNOT

|0> |0> iYiY -iY

FIGURE B.1: Implementation of a Toffoli gate using an ancilla qubit, a CNOT and controlled-controlled-iY and controlled-controlled-(iY)† gates.

In this Appendix we show how the Toffoli gate can be constructed using Hadamard, CNOT and controlled-Hadamard gates with the help of an ancilla qubit initialized in |0i. The key to this construction is shown in Fig. B.1, where the Toffoli gate is obtained using a CNOT gate, a controlled-controlled-iY and its hermitian conjugate. The goal is to show that the controlled-controlled-iY, and its Hermitian conjugate can be obtained from Hadamard, CNOT and controlled-Hadamard. To this end we follow the general construction of a controlled-controlled-U described in Sec. 4.3 of Ref. [4]. In order to use this method to construct the controlled-controlled-iY operator, we have to ﬁnd a unitary operator V such that V2 = iY, which turns out to be V = ZH. As shown in Fig. B.2 the controlled-controlled-iY is readily obtained in terms of CNOT, controlled-Hadamard and controlled-Z gate (CZ). Finally, using the well-known relation between the CZ gate and the CNOT gate shown in Fig. B.3, we complete the synthesis of a Toffoli gate using Hadamard, CNOT and controlled- Hadamard, showing the universality of these sets of quantum gates.

iY Z H H Z HZ HZ

FIGURE B.2: Implementation of a controlled-controlled-iY gate using CNOT, controlled-Hadamard and controlled-Z gates. The implementation of controlled-controlled-(iY)† follows analogously. 156 Appendix B. Universality of Hadamard, controlled-Hadamard and CNOT

Z H H

FIGURE B.3: Controlled-Z gate from Hadamard and CNOT. 157

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