In Situ Quantum Control Over Superconducting Qubits

In situ quantum control over superconducting qubits

Anatoly Kulikov M.Sc.

A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2020 School of Mathematics and Physics ARC Centre of Excellence for Engineered Quantum Systems (EQuS)

ABSTRACT

In the last decade, quantum information processing has transformed from a field of mostly academic research to an applied engineering subfield with many commercial companies an- nouncing strategies to achieve quantum advantage and construct a useful universal quantum computer. Continuing efforts to improve qubit lifetime, control techniques, materials and fabrication methods together with exploring ways to scale up the architecture have culminated in the recent achievement of quantum supremacy using a programmable superconducting processor – a major milestone in quantum computing en route to useful devices. Marking the point when for the first time a quantum processor can outperform the best classical supercomputer, it heralds a new era in computer science, technology and information processing. One of the key developments enabling this transition to happen is the ability to exert more precise control over quantum bits and the ability to detect and mitigate control errors and imperfections. In this thesis, ways to efficiently control superconducting qubits are explored from the experimental viewpoint. We introduce a state-of-the-art experimental machinery enabling one to perform one- and two-qubit gates focusing on the technical aspect and outlining some guidelines for its efficient operation. We describe the software stack from the time alignment of control pulses and triggers to the data processing organisation. We then bring in the standard qubit manipulation and readout methods and proceed to describe some of the more advanced optimal control and calibration techniques. The described toolbox together with some newly developed advances such as multi-level single-shot readout are applied to a practical problem of generating high-quality random numbers. Following the recent advances in the field, we certify our generator based on the Kochen- Specker theorem utilizing contextuality of quantum mechanics as an underlying resource of randomness. Our implementation based on the solid-state circuit quantum electrodynamics (QED) platform and the new certification scheme allows us to greatly enhance the bit rate of certified quantum random number generators (QRNGs) and approach the requirements for real-world applications. Addressing some of the control and state preparation errors, we develop a set of techniques to use a qubit as an in situ probe of distortions of its control lines and its effective temperature. Considering a qubit simultaneously driven by a longitudinal and a transverse MW-drives under certain resonant conditions, we show that in the rotating dressed-state picture (second rotating frame) one can get information about both the amplitude and the phase of the drive signals. This allows us to employ a qubit as an on-chip vector network analyzer and precisely measure the transfer functions of its control lines. As no other method with comparable frequency domain coverage exists, we verify our technique by cancelling out the distortions and increasing the

i ﬁdelity of entangling two-qubit CPHASE gate. Finally, we show that the decay of the correlations between two sequential measurements of a quantum bit is a manifestation of its nonzero eﬀective temperature. We use it to develop a precise method of evaluating the qubits population allowing for virtually unlimited suppression of absolute errors. We realize the method and demonstrate its applicability to the calibration of a quantum processor and evaluation of active reset techniques. As the actual mechanism of the increased excited-state population is currently unknown, we use our method’s precision to show the qubit’s coupling to two separate independent thermal baths. We also introduce the ‘temperature spectroscopy’ as a tool to acquire more information about the qubits spurious excitation sources.

ii DECLARATION BY AUTHOR

This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including development of concepts, data analysis and post-processing, experimental techniques, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis.

iii PUBLICATIONS DURING CANDIDATURE

Peer-reviewed publications

1. Measuring eﬀective temperatures of qubits using correlations, A. Kulikov, R. Navarathna, and A. Fedorov Phys. Rev. Lett. 124 240501 (2020)

2. In situ Characterization of Qubit Control Lines: A Qubit as a Vector Network Analyzer, M. Jerger, A. Kulikov, Z. Vasselin, and A. Fedorov Phys. Rev. Lett. 123, 150501 (2019)

3. On the chromatic number of small-dimensional Euclidean spaces, D. Cherkashin, A. Kulikov, and A. Raigorodskii Discrete Applied Mathematics 243, 125-131 (2018)

4. Realization of a Quantum Random Number Generator Certiﬁed with the Kochen-Specker Theorem, A. Kulikov, M. Jerger, A. Potoˇcnik,A. Wallraﬀ, and A. Fedorov Phys. Rev. Lett. 119, 240501 (2017)

iv PUBLICATIONS INCLUDED IN THIS THESIS

Realization of a Quantum Random Number Generator Certiﬁed with the Kochen-Specker Theorem, A. Kulikov, M. Jerger, A. Potoˇcnik, A. Wallraﬀ, and A. Fedorov Phys. Rev. Lett. 119, 240501 (2017). (Included in Chapter4.)

Contributor Statement of contribution A. Kulikov Planning the experiment (Candidate) Conducting the measurements Analysis of the data Writing the manuscript Referee replies and final manuscript revision M. Jerger Building the measurement setup Assisting with the measurements Co-supervising the project A. Potoˇcnik Fabricating the parametric amplifier A. Wallraff A. Fedorov Supervising the project Fabricating the qubit Writing the manuscript

In situ Characterization of Qubit Control Lines: A Qubit as a Vector Network Analyzer, M. Jerger, A. Kulikov, Z. Vasselin, and A. Fedorov Phys. Rev. Lett. 123, 150501 (2019). (Included in Chapter5.)

v Contributor Statement of contribution A. Kulikov Developing the software (Candidate) Conducting the measurements (gate optimisation) Analysis of the data (gate optimisation) Writing the ﬁnal manuscript Referee replies and ﬁnal manuscript revision M. Jerger Developing the theoretical proposal Planning the experiment Developing the software Conducting the measurements (line calibration) Analysis of the data (line calibration) Writing the initial manuscript Z. Vasselin Performing the initial line calibration measurements

A. Fedorov Supervising the project Fabricating the qubit Developing the theoretical protocol Writing the ﬁnal manuscript Referee replies and ﬁnal manuscript revision

Measuring eﬀective temperatures of qubits using correlations, A. Kulikov, R. Navarathna, and A. Fedorov Phys. Rev. Lett. 124 240501 (2020). (Included in Chapter6.)

Contributor Statement of contribution A. Kulikov Planning the experiment (Candidate) Conducting the measurements Analysis of the data Writing the manuscript Referee replies and final manuscript revision R. Navarathna Conducting the measurements Analysis of the data Writing the manuscript Referee replies and final manuscript revision A. Fedorov Supervising the project Writing the manuscript Referee replies and final manuscript revision

vi CONTRIBUTIONS BY OTHERS TO THE THESIS

No contributions by others except as co-authors as outlined in the “Publications included in this thesis” section.

vii STATEMENT OF PARTS OF THE THESIS SUBMITTED TO QUALIFY FOR THE AWARD OF ANOTHER DEGREE

No work submitted towards another degree have been included in this thesis.

viii RESEARCH INVOLVING HUMAN OR ANIMAL SUBJECTS

No animal or human subjects were involved in this research.

ix ACKNOWLEDGEMENTS

First and foremost, I would like to thank my principal advisor Arkady Fedorov for introducing me to the exciting field of quantum technology, and for the invitation to pursue this PhD project in the SQD Lab in Australia. Your calm and focused work attitude together with clear understanding of theoretical and experimental concepts have taught me a great deal. Further, a perfect combination of guidance and freedom, and being always welcome to discuss ideas have made working with you very productive and enjoyable. The technical part of this work would not have been possible without the extensive help of my associate advisor Markus Jerger. Markus knows every device and every line of code in the lab, and was always happy to share his knowledge and help solving any arising complications, which we had no shortage of. I enjoyed very much our technical discussions, as well as working together on the projects. I would like to express my deep gratitude to all members of the SQD group for the wel- coming and cheerful atmosphere, and for a true feeling of working in a team. AndrésRosario was an amazing colleague to work with. Thank you for your great humour, positive approach to everything in the lab, interesting discussions and for persuading me to bring some superconductivity into our everyday life. Volodymyr Monarkha knew how to fix any device without even reading a manual (an old car, too!), and his intuition and help with hands-on parts of lab work were invaluable. It was very curious to explore the limitations of our hardware and software with Rohit Navarathna and Alejandro Gomez. Thank you guys for your help and nice times! Eugene Sachkou’s logical and coherent approach to work, and always willing to give a hand have been of great aid during the later stages of my PhD. Daniel Szombati, Jovian De- laforce, Pradeep Kumar, and most recently Tina Moghaddam, Eric Xin He and Rohit Beriwal have been superb to work with and explore all sorts of experimental and theoretical questions. Finally, I would like to thank the group for being good friends, and for fun times in and out of work. Daily lunch discussions, hiking and snorkelling trips and get-togethers made my PhD life very colourful and enjoyable. It was a true pleasure to work with each and everyone of you. Apart from being a delightful friend, Christina Giarmatzi has shared heaps of insight and wisdom about a PhD student’s life, for which I am very grateful. Spyros Tserkis was a great person to have all sorts of discussions with, and his bold and original way of thinking has made talking to him truly interesting and enjoyable. My officemates and colleagues have filled the work days with an optimal mix of focused concentration and cheering laughter. Thank you Alejandro Gomez, Sarah Lau, Behnam Tonekaboni, Nick Wyatt, Dat Thanh Le, Ming Su, Leo Morais, Rob Harris! I would like to thank people of the Engineered Quantum Systems centre – Andrew White, Marcelo P. de Almeida, Magdalena Zych, Tom Stace, Josh Combes, Clemens Müller,Gerard

x Milburn and many others for their help and guidance, casual physics discussions at coffee breaks and for sharing their curiosity and passion for so many aspects of quantum world. My special gratitude goes to the administrative staff of EQuS – Angela Bird, Lisa Walker, Tara Massingham and Joyce Wang – for their help with formal and financial matters, and especially for organising a series of amazing events for the centre. Murray Kane was very helpful with the administrative part of my PhD, and a very interesting person to have philosophical discussions with. Finally, I would like to thank Arkady, Eugene and Alejandro for careful proof-reading of this thesis and for suggesting numerous ways to improve it.

xi FINANCIAL SUPPORT

This research was supported by:

• the University of Queensland International Scholarship (tuituion fee award and living allowance stipend), awarded by The University of Queensland

• the Research Higher Degree Scholarship (living allowance top-up), awarded by the Aus- tralian Research Council Centre of Excellence for Engineered Quantum Systems (CE110001013 and CE170100009)

xii KEYWORDS

Keywords: circuit QED, superconducting qubits, quantum mechanics, quantum information, quantum control, microwave engineering, randomness generation, quantum computing.

xiii AUSTRALIAN AND NEW ZEALAND STANDARD RESEARCH CLASSIFICATIONS (ANZSRC)

ANZSRC code: 020603 Quantum Information, Computation and Communication 60% ANZSRC code: 090699 Electrical and Electronic Engineering not elsewhere classiﬁed, 20% ANZSRC code: 020604 Quantum Optics, 20%

xiv FIELDS OF RESEARCH (FOR) CLASSIFICATION

FoR code: 0206 Quantum Physics, 60% FoR code: 0202 Atomic, Molecular, Nuclear, Particle and Plasma Physics, 20% FoR code: 0906 Electrical and Electronic Engineering, 20%

xv xvi CONTENTS

Abstract...... i Publications during candidature...... iv Acknowledgements...... x List of Figures...... xxi List of Tables...... xxix

1 Introduction1 1.1 Quantum information processing...... 1 1.2 Superconducting circuits for QIP...... 2 1.3 This thesis...... 3 1.4 Brief chapter description...... 4

2 Artiﬁcial quantum systems and superconducting qubits 11 2.1 Natural qubits...... 11 2.2 Artiﬁcial quantum systems...... 12 2.3 Superconducting LC circuit...... 13 2.3.1 Superconducting qubit...... 14 2.4 Transmon qubit...... 16 2.5 dc-SQUID and tunable transmon qubit...... 18

3 Experimental toolbox 23 3.1 General requirements...... 23 3.2 Experimental setup...... 24 3.2.1 Room-temperature setup...... 24 3.2.2 Cryostat wiring...... 28 3.2.3 Mixing chamber plate setup...... 29 3.2.4 Qubit controls...... 32 3.2.5 Output signal amplification chain...... 34 3.2.6 Josephson parametric amplifier...... 34 3.3 Spectroscopy & calibration measurements...... 35 3.3.1 VNA...... 35 3.3.2 Resonator spectroscopy...... 35 3.3.3 Qubit spectroscopy...... 38 3.3.4 Spectroscopy of the higher transitions...... 39 3.3.5 Dispersive shift...... 42 3.3.6 Resonator spectroscopy vs flux...... 43

xvii 3.3.7 Qubit spectroscopy vs ﬂux...... 44 3.4 Time-domain measurements...... 46 3.4.1 Pulse alignment organization...... 47 3.4.2 Instrument delays and virtual timing instrument...... 49 3.4.3 PulseGen...... 50 3.4.4 Readout in time domain...... 51 3.4.5 Averaging and normalization...... 52 3.4.6 Single shot readout...... 54 3.4.7 Three-level single shot readout...... 57 3.4.8 Rabi measurement...... 58 3.4.9 Ramsey experiment...... 60 3.4.10 Rabi chevron...... 62

3.4.11 T1 and T2 times...... 63 3.4.12 Qubit active reset...... 65

4 Quantum Random Generator Certiﬁed with the Kochen-Specker Theorem 73 4.1 Certiﬁed QRNGs...... 73 4.2 Protocol and realization...... 74 4.2.1 Three-level single-shot readout...... 77 4.2.2 Robustness to noise and imperfections...... 77 4.3 Discussions...... 78

5 In situ characterization of qubit control lines: a qubit as a vector network analyzer 83 5.1 Introduction...... 84 5.2 Theory...... 85 5.2.1 Non-resonant driving case...... 87 5.3 Experimental procedure...... 88 5.4 Detailed experimental procedure...... 89 5.5 Realization...... 91 5.6 Quality of the phase reconstruction...... 95 5.7 Reconstruction of the transfer function of the ’charge’ (transverse) control line. 96 5.8 Fundamental accuracy limits...... 96 5.8.1 Rabi frequency change due to the RW approximation...... 97 5.8.2 Error of reconstruction in the ﬁrst rotating frame...... 97 5.8.3 Error of reconstruction in the second rotating frame...... 98 5.9 Error analysis...... 98 5.10 Acknowledgements...... 101

6 Measuring eﬀective temperatures of qubits using correlations 105 6.1 Experimental platform...... 106 6.2 Theoretical idea...... 107

xviii 6.2.1 Approximate expression derivation...... 109 6.2.2 Normalizing the responses...... 109 6.2.3 Added measurements noise...... 111 6.3 Results...... 111 6.3.1 Relaxation time vs Frequency...... 114 6.4 Precision and errors...... 115 6.4.1 Direct counting...... 115 6.4.2 Qutrit...... 116 6.4.3 Correlator method...... 117 6.4.4 Slow noise contributions...... 118 6.5 Discussions...... 119

xix xx LIST OF FIGURES

2.1 An atom as a natural qubit (qutrit)...... 12

2.2 An ideal LC-oscillator...... 13

2.3 Harmonic spectrum of an LC-oscillator...... 14

2.4 A non-linear LC-oscillator. Marked with a cross is a Josephson junction..... 15

2.5 Anharmonic spectrum of a nonlinear system comprised of a capacitor and a Josephson junction...... 16

2.6 Typical energy spectrum of a transmon qubit. First three levels allow operating it as a qubit (only the ﬁrst transition), or a qutrit – quantum three-level system, both transitions...... 17

2.7 SQUID loop consisting of two Josephson junctions with external magnetic ﬂux applied...... 18

2.8 Full scheme of a tunable transmon qubit. A single Josephson junction is replaced by a loop containing two junctions, which makes the qubit frequency in-situ tunable by the external magnetic ﬂux. The loop is shunted by a large capacitor

Cg leading to a small charging energy EC . Parasitic capacitances of the junctions are shown for the completeness of the model...... 19

3.1 High-frequency local oscillator signal coming from a stable microwave source is combined on an I-Q mixer with a low-frequency shaped pulsed signal coming from an AWG, resulting in a pulsed high-frequency signal required to drive a qubit. Each setup (microwave source + I-Q mixer + AWG) is capable of driving multiple transitions of one qubit...... 26

3.2 An FPGA accepts a frequency reference clock input, two data inputs (ADC 1 and 2), and an acquisition trigger input. Math module is optional and allows performing amplitude, power or correlation measurements between two FPGA channels. This Figure is adopted from [16]. For more details on principles of FPGA operation, programming, connections and descriptions of internal modules see Appendix A in [16]...... 27

3.3 The output signal is routed to a HEMT ampliﬁer via a series of circulators to stop the thermal noise from propagating back...... 30

xxi 3.4 Output signal passes a circulator for isolation purposes and then routed to a directional coupler (DC), where a strong pump tone to activate the Josephson parametric amplifier (JPA) is added. The signal then continues through to the JPA, where it is amplified and reflected. The amplified signal is routed upwards to another circulator for isolation and continues towards an output line and a HEMT amplifier...... 31 3.5 Full simplified diagram of a measurement setup used in Chapter4. A transmon type multi-level quantum system is incorporated into a 3D microwave copper cavity attached to the cold stage of a dilution cryostat. A magnetically tunable Josephson junction loop (SQUID, see Sec. 2.5) is used to control the transition frequency of the qutrit by a superconducting coil attached to the cavity. Amplitude-controlled and phase-controlled microwave pulses are applied to the input port of the cavity by a quadrature IF (IQ) mixer driven by a local oscillator (LO) and sideband modulated by an arbitrary waveform generator (AWG). The measurement signals transmitted through the cavity are amplified by quantum Josephson parmateric amplifier (JPA), by a high-electron-mobility transistor (HEMT) amplifier at 4 K and a chain of room temperature (RT) amplifiers. JPA is also magnetically tunable allowing to match its amplification range with the cavity output (readout) frequency. The sample at 20 mK is isolated from the higher temperature stages by a series of circulators. The amplified transmission signal is down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated LO, and is digitized by an analog-to-digital converter (ADC) for data analysis...... 33 3.6 Resonator spectroscopy measurement fit to a positive Lorentzian shape. Abso- lute value of transmitted voltage is plotted on the y-axis...... 36 3.7 Resonator spectroscopy as a function of the probe tone source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. At sufficiently high power (≈ −35 dBm at the source) the resonator peak start getting distorted and is sharply replaced by a bright mode peak at around - 12 dBm...... 37 3.8 Qubit spectroscopy measurements. Absolute value of transmitted voltage is plotted on the y-axis. (a) Qubit spectroscopy measurement fit to a negative Lorentzian shape. Low-power regime ensures a single narrow dip (b) High applied power showcases the dispersive shift of the qubit frequency, thus resulting in two dips corresponding to zero and one photon in the resonator...... 38 3.9 Qubit spectroscopy measurement with too high resonator power...... 39 3.10 Qubit spectroscopy as a function of the resonator source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. The qubit dip broadens and shifts to lower frequency as resonator is populated with more than one photon on average at ≈ −45 dBm power at the source.... 40

xxii 3.11 Qubit spectroscopy as a function of the qubit probe source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. As the qubit power increases, the qubit dip get wider and the second dip corresponding to one photon at the resonator appears (compare to Fig. 3.8b). With even higher power two-photon Sg⟩-Sf⟩ transitions are visible...... 41 3.12 Resonator spectroscopy with a driven qubit transition(s). (a) Qubit’s Sg⟩-Se⟩ transition is driven, thus we observe two peaks. (b) Ax extra microwave source is used to continuously drive an Se⟩-Sf⟩ transition as well. The three resonator peaks are labelled with respect to the corresponding qubit state...... 42 3.13 Measuring resonator spectroscopy as a function of the voltage applied to the qubit bias coils. (a) Modulation of the resonator frequency due to the qubit- resonator coupling. (b) An avoided crossing between the resonator and the qubit. One can directly determine the qubit-resonator coupling g by observing this plot...... 43 3.14 Qubit spectroscopies at different voltages on the magnetic coil DC source. Each vertical slice corresponds to a single qubit spectroscopy normalized to a maximum transmission, so only qubit dip(s) are visible. (a) Two qubits’ spectroscopies are performed in the same measurement, demonstrating a different qubits’ coupling to the coil DC source. Note qubit 1 is periodic in flux having maxima corresponding to 0, 1 or two flux quanta through the SQUID loop (b) A demonstration of a spectroscopy with a sliding window for a single qubit... 45 3.15 Principal scheme of the devices required to run a multi-qubit experiment, see details below in the text. Black arrows indicate clock synchronization signals, i.e. stable 10 MHz tone (1 GHz tone between the microwave sources, if the model allows). Red arrows correspond to timing triggers. Note that FPGAs could be triggered either by the pulser, or by AWGs depending on the choice of the wiring. 47 3.16 Principal timing scheme for a single experiment repetition...... 48 3.17 Principal timing scheme for a single experiment repetition including triggering pulses and markers. Internal trigger delays of instruments and natural delays due to wiring are not taken into account on this schematic picture...... 49

3.18 This ﬁgure demonstrates adding a new pulse P4 to the set of three pulses P1, P2

and P3 in the pulse generating software. Notice how even though the pulse is added to the end of the sequence, it affects the timings of all other pulses, as the last pulse always ends at the fixed point (see Sec. 3.4.1)...... 51 3.19 I-Q plane hexbin histogram plots of averaged readout for a qubit prepared in three different states. The outcomes are colour-coded. The intensity of the colour represents the number of measurement outcomes falling in each bin. Each measurement point is averaged with 16 (a) and 1024 (b) repetitions. Note that while averaging clearly helps reducing the noise, the single measurement features of a π~2 state also get averaged out...... 53

xxiii 3.20 I-Q plane hexbin histogram plot of single shot readout for a qubit. The outcomes are colour-coded with blue corresponding to the qubit prepared in the ground state, and red – in the excited. The intensity of the colour represents the number of measurement outcomes falling in each bin. Unlike the averaged case, each single point is a result of a single measurement, high contrast of which allows distinguishing the outcomes...... 55

3.21 I-Q plane hexbin histogram plot of single shot readout for a qubit. (a) Reference plot with the qubit being prepared either in Sg⟩ or Se⟩ state. (b) The qubit is prepared in the superposition π~2 state. Note the key diﬀerence from 3.19a and 3.19b: the superposition state does appear as two clouds of points, reproducing the probabilistic outcomes of being projected either to Sg⟩ or to Se⟩...... 57

3.22 I-Q plane hexbin histogram plots of single shot three-level readout for a qutrit. The outcomes are colour-coded. The intensity of the colour represents the number of measurement outcomes falling in each bin. (a) The readout frequency equals to the resonator’s peak corresponding to the Sg⟩ state of a qutrit, i.e. a standard readout. Note the contrast between the Se⟩ and Sf⟩ states is virtually absent. (b) The readout frequency is oﬀset by 7.4 MHz and is closer to the Se⟩-state resonator frequency, thus leading to much better contrast between Se⟩ and Sf⟩. Both readouts are not optimised with respect to the readout length and pulse power, thus one can see a lot of decay during readout (especially clear on 3.22b)...... 58

3.24 The experimental protocol for Rabi and Ramsey experiments. Each protocol is repeated multiple times with various magnitudes of the applied pulse α (Rabi) or with various delays τ between consecutive π~2 pulses...... 59

3.23 Rabi oscillations on a Bloch sphere. The rotation vector Ω depends on the phase of the drive, thus allowing us to apply diﬀerent gates...... 59

3.25 Rabi ﬂopping experiment. Fitting to an oscillation function yields an amplitude for a π and π~2 pulses...... 60

3.26 Two applications of Ramsey sequence. (a) Allows to calibrate the qubit frequency with high precision. We add extra detuning to make ﬁtting more robust and decrease the length of the delay τ between the π~2-pulses. (b) Longer Ram- sey sequence allows observing the eﬀects of qubit’s decoherence, see more details in Sec. 3.4.11...... 61

3.27 A theoretical modelling of the Rabi chevron pattern. The colorbar corresponds to the probability to ﬁnd the qubit in an excited state. Detuning of the drive frequency (x-axis) from the qubit’s energy splitting leads to faster Rabi oscillations yet being tilted on the Bloch sphere, thus not reaching the Se⟩ state fully...... 63

xxiv 3.28 The experimental protocols for T1 and T2 measurements. (T1) For each segment a π pulse is applied, followed by a waiting time τ before taking a measurement. To maximize the measurement precision, τ is to be swept from zero times to

several (3-4) T1s. (T2) While the protocol is identical to the Ramsey sequence

outlined in Sec. 3.4.9, the detuning of the pi-half pulses in T2 measurement is smaller and the waiting times τ are swept in much larger range to observe the decay of the Ramsey fringes...... 64

3.29 Typical plots for a T1 and T2 measurements. On the y-axis for both plots is the

σz expectation value...... 64

3.30 Calibration sequence for a reset pulse. (a) Pulsed spectroscopy of the reset transition. On the x-axis is the intermediate frequency supplied to the IQ mixer by an AWG. (b) Rabi-like damped oscillations allow determining the length of the reset π pulse...... 66

3.31 Single shot readout of a qubit prepared in the ground state. The repetition time of 6.4 µs is less than the relaxation time of the qubit. (a) No reset sequence is applied. A considerable amount of time the qubit is measured to be in the Se⟩ state. (b) An application of the reset sequence greatly enhances the quality of initialization...... 67

4.1 The theoretical protocol of the QRNG certiﬁed by the strong Kochen-Specker theorem proposed in Ref. [13]. The protocol is formulated for a spin-1 particle and consists of two sequential measurements. The ﬁrst measurement is used to

initialize the particle in the Sz = 0 eigenstate of the spin operator Sz. The second

measurement is performed in the eigenbasis of the Sx operator with the two

outcomes Sx = ±1 realized randomly as proven by the Kochen-Specher theorem.

The outcome Sx = 0 is never realized in the ideal case but can be used to monitor the quality of the protocol implementation...... 74

xxv 4.2 (a) Simplified diagram of the measurement setup. A transmon type multi-level quantum system is incorporated into a 3D microwave copper cavity attached to the cold stage of a dilution cryostat. A magnetically tunable Josephson junction (SQUID) is used to control the transition frequency of the qutrit by a superconducting coil attached to the cavity. Amplitude-controlled and phase-controlled microwave pulses are applied to the input port of the cavity by a quadrature IF (IQ) mixer driven by a local oscillator (LO) and sideband modulated by an arbitrary waveform generator (AWG). The measurement signals transmitted through the cavity are amplified by quantum Josephson parmateric amplifier (JPA), by a high-electron-mobility transistor (HEMT) amplifier at 4 K and a chain of room temperature (RT) amplifiers. The sample at 20 mK is isolated from the higher temperature stages by three circulators (C) in series. The amplified transmission signal is down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated LO, and is digitized by an analog-to-digital converter (ADC) for data analysis. (b) The energy level diagram of a qutrit coupled to a microwave cavity. The transition frequencies of the qutrit and cavity are in GHz while the anharmonicity of the qutrit is ∼ 300 MHz. When the coupling g between the transmon and the cavity is much smaller than their mutual detuning, the system is in the dispersive regime used for measurement of the qutrit...... 76

4.3 (a) Hexbin histogram plot of single-shot three-level readout of diﬀerent qutrit states. Red: ground qutrit state; Green – excited state (S1⟩). Blue: second excited state (S2⟩). The intensity of the color represents the number of measurement outcomes falling in each bin. (b) Hexbin plot of the output of the protocol. Shown are logical encoding of the resulting states and the correspondence to the

spin-1 protocol. Note, the Sx = 0 state is almost (< 0.1%) never realized. The black lines sketch the boundaries of the classiﬁcation regions...... 78

5.1 (Top) Prepared qubit pulse is being distorted by the control line. Resulting signal is the convolution (in time domain) of the programmed pulse and the impulse response of the control line. (Bottom) Knowing the transfer function G(ω) or an impulse response g(t) allows applying its inverse to the input in pre-processing and feeding the resulting pulse through the control line...... 85

5.2 Left: Rabi oscillations in the ﬁrst rotating frame. Right: Eﬀective Rabi oscillations of a dressed qubit in the second rotating frame...... 86

xxvi 5.3 A typical data set for ΩR = 19.8 MHz and the corresponding trajectories of the Bloch vector in the ﬁrst rotating frame with only x drive applied, with both x and z drives and with both x and z drives in the second rotating frame. a) State tomography of the qubit with only the x drive applied. The ﬁt yields a rotation ⃗ vector of Θ=(19.7,-1.9,0.8) MHz and φx = −0.1. b) State tomography of the qubit

with both the x and z drives applied. The frequency of z drive is set to ωz = ΩR. The sampling rate of the experiment is chosen such that the slow oscillations

at ωR can be resolved unambiguously but not necessarily the fast oscillations at

ΩR, which are canceled by U3. The orange curve shows the theoretical dynamics derived from the ﬁt in (c). c) The dynamics of the qubit in the second rotating frame, transformed from (b). The ﬁt yields θ⃗=(0.2,-3.5,0.1) MHz and we can

obtain Az = 3.51 MHz and φz = −1.54...... 89

5.4 Amplitudes Az and phases φz of the transfer functions of the z control line. Points in blue and green show the response of the flux line for two different configurations (see text for details). Darker points were measured with resonant

driving ωx = ωq by varying Ax, lighter points were measured by varying ωx − ωq

with ﬁxed Ax (oﬀ-resonant case)...... 91

5.5 Amplitude and phase of transmission through a transmission line with a shorted

stub resonator. Points in blue were measured with the qubit by comparing Az

and φz with and without the stub in place, the orange line was measured directly with a commercial vector network analyzer...... 92

5.6 Vaccum Rabi oscillation between the S11⟩ and S20⟩ states of two Transmon qubits assuming (a) a perfect impulse response, (b) the impulse response measured at room temperature and (c) the impulse response measured using the qubit... 93

5.7 Quality of the phase reconstruction – (top) Measured y and z components of the Rotation vector of the dressed qubit as the phase of the longitudinal driving pulse programmed at the AWG is varied. (bottom) Diﬀerence between the phase

φz extracted by the method and the phase programmed at the AWG...... 95 ⃗ 5.8 Amplitude Ax and rotation vector Θ reconstruction – Rotation of the bloch vector of a transmon circuit compared to a two-level system. With increasing driving amplitudes, the Rabi frequency of the transmon is decreased compared to the qubit and a spontaneous detuning can be observed...... 99

5.9 Relative error in the Az reconstruction as a function of frequency. Green and red curves are calculated taking into account decay and decoherence, but assuming a perfect qubit, i.e. the system is truncated to two levels. Green curve corresponds to resonant driving and red is to the off-resonant one. Blue and orange curves (resonant and off-resonant, respectively) are constructed from a full Transmon simulation with the Transmon truncated to the first ten levels...... 100

xxvii 5.10 Error in the φz reconstruction as a function of frequency. Color-coding matches the plot above. One may note that even for a two-level case with higher detunings phase reconstructions gets less precise due to emergence of ﬁtting errors due to low signal contrast...... 101

6.1 Decay of the normalized correlator between two sequential measurements sep-

arated by τ. The solid lines represent exponential T1-decay. Amplitude of the correlator at zero (or lowest attainable) delay allows one to reconstruct the Se⟩- state population and hence the eﬀective temperature of the qubit...... 108 6.2 The experimental protocol. “Run I” represents measurement of the correlation function g(1)(τ) and g(0). “Run II” is an additional calibration measurement

required for correct scaling of Pe. The variable delay was used to measure the (1) decay of g (τ). To determine Pe only one measurement with τ = 0 is necessary. 108 6.3 Single-shot measurement data with no pulse (π pulse) shown as blue (red) pixels.

The white (black) circle shows the averaged response V˜g (V˜e). Dashed is the line

joining Vg and Ve; We assume deviations perpendicular to this line are only caused by noise, and project the complex IQ measurement onto it...... 110 6.4 (a) Measured Se⟩-state population as a function of the mixing chamber sensor temperature. Red points are correlator measurements with a JPA. Data for each point corresponds to 220 repetitions. The blue points are measured with JPA turned off. The black solid line corresponds to the M-B distribution offset by 0.33% as indicated by the dashed green line. The error-bars cover two standard deviations in measurement (95% confidence). (b) Deviation of the data from the M-B distribution for different methods (see text for more details)...... 112 6.5 Excited state population vs qubit frequency representing a “noise spectrum” as seen by the qubit. The green arrow indicates the qubit frequency used for the rest of the experiments...... 113 6.6 a) Excited state population and b) Relaxation time vs qubit frequency. The green arrow shows the frequency used in the rest of the experiments. The black dashed arrows indicate the simultaneous jumps in population and relaxation time, indicating that they are related. The purple line is the Purcell limit of relaxation time...... 114

6.7 Relative precision of Pe measurements and their linear ﬁts. The precision scales as expected for uncorrelated noise (indicated by the black dashed line). Inset: Population (solid line) and standard deviation (ﬁll) for a measurement power of −30 dBm...... 118

xxviii LIST OF TABLES

xxix CHAPTER 1

INTRODUCTION

1.1 Quantum information processing

It all started with an idea conventionally attributed to Richard Feynman [1], to simulate complex quantum systems using simpler and controllable ones. A decade later, in early 90’s, a new field named quantum information processing (QIP) got a major thrust due to the works of Peter Shor [2] and Lov Grover [3], who proposed quantum algorithms which possibly contradicted the Church-Turing thesis and used another computational paradigm, not covered by the classical Turing machine [4]. Naturally, potential applications turned a mere theoretical curiosity into a question well worth considering and heated the debate if it was possible to realize such a quantum simulator [5]. A theoretical proposal of an experimental platform followed soon after, with the idea to encode a quantum bit into states of a trapped ion and an outline how one- and two-qubit gates could be realized [6]. Successful experimental realization of the proposal [7] finally established a new field as a promising research area. Quantum information processing has continued its development by broadening the scope of applications it could potentially tackle, and by expanding to other physical platforms to realize the theoretical ideas on [8–10]. Emerged independently, the domain of quantum optics allows using photons as flying qubits with quantum information encoded in polarization, frequency, time of arrival or even the shape of photons [9]. Fast propagating speeds, relative ease of manipulation by existing technology and in-place infrastructure make optical qubits to fit naturally as carriers of information, driving the progress of quantum communication and quantum key distribution fields. With a tremendous progress through the last two decades, recent highlights in the area include quantum key distribution between a satellite and a ground-based laboratory [11], prospects of creating a secure quantum internet [12, 13] and a list of commercial companies using the emerging technology to offer ”quantum-safe network encryption” [14]. While for quantum communication the reign of quantum optics in indisputable, a plethora of hardware platforms and qubit modalities have been proposed for quantum computing purposes. Long coherence times, achieved excellent fidelities of single- and two-qubit gates and readout have ensured various types of trapped-ion qubits to remain as one of the leading platforms, particularly for quantum simulation [10, 15]. Spin qubits offer even higher fidelities of the universal gate set operations, though suffer from more complicated scaling to higher numbers of qubits. Somewhat different from natural spins and ions, superconducting qubits

1 CHAPTER 1. INTRODUCTION emerged in 1999 [8, 16] are based on Josephson junctions, artificial in nature and mesoscopic in size [17, 18]. Superconducting circuits and qubits leverage well-established nanofabrication techniques, developed microwave machinery and – above all – easier compared to other systems scaling, which gradually made them a workhorse for quantum computing throughout the community [19–21]. Finally, many other qubit systems are explored in laboratories throughout the world to potentially supersede the existing platforms. As a part of an even broader field of quantum technologies, quantum information processing nowadays attracts a lot of interest and increased funding from research and governmental agencies. While the particular attention is directed to the potential impacts on cryptogra- phy and communication security, the focus is now shifting towards the applications the noisy intermediate-scale quantum (NISQ) [22] devices have to offer [23–25].

1.2 Superconducting circuits for QIP

Having its roots in 1950s, the fundamental physical research field of cavity quantum electrodynamics (cavity QED) was explored from both theoretical and experimental grounds. Motivated by the study of light-matter interaction in a simpler, controlled environment, it was examining coupling and interactions of single quantum systems to single light modes [26, 27]. Cavity QED offers a natural way to modify the environment seen by an atom and significantly increase (or decrease) the coupling to a specific light mode by embedding the atom into an optical cavity. The first results in the field were concerned with spontaneous emission of a single spin or an atom. Depending on whether the cavity is resonant or far off-resonant to the atom, one can greatly enhance [28] or suppress [29] the rate of atom’s spontaneous emission and thereby modify the lifetime of the excited state. Coupling of the atomic states to the vacuum field within the cavity shifts the transition energy levels of the embedded atom [30], which can be observed by performing simple spectroscopic measurements. Finally, reaching the strong coupling regime (where the coupling exceeds loss rates) allows exchanging the quantum information between the light mode and the atom [31], giving a certain degree of control over the properties of a joint cavity-atom system and allowing to observe a range of effects predicted by quantum theory [27]. This was sought to address one of the main challenges in the experimental QIP: creating a system well decoupled from the environment so the coherence times are high, but also well coupled to the control signals and pulses [26]. Applying the ideas of cavity QED to superconducting circuits and qubits led to advent of circuit QED in 2004 [32], a platform that continues to establish itself as the most promising technology to achieve robust and scalable quantum computers [17, 33], and which this thesis is based on. The success of the platform is highlighted by recent advances by commercial companies – Google Sycamore chip with 53 superconducting qubits [20] used to demonstrate quantum supremacy [34] and IBM cloud quantum computing initiative [19] with multiple quantum processors available and accessible to general public and scientific community to perform experiments, with the latest processor also containing 53 qubits [35]. While these devices are

2 1.3. THIS THESIS still far from the ﬁrst useful commercial application, and many technological challenges are yet to be solved, they readily demonstrate the reliability and scalability of the superconducting qubits and circuit QED.

1.3 This thesis

One of the main goals of the quantum information processing part of experimental physics is to transfer manipulations with quantum systems from the language of physics to the language of abstractions. In the same way as in electronics we depart from real physical objects to the lumped element abstraction of resistors, capacitors and inductances, then move on to creating elements which lie within the bounds of our abstractions and get ability to make complex and scalable digital circuits, we would like to depart from real physical systems to the language of qubits and gates. A crucial step in such a transition for circuit QED would be attaining exquisite control over superconducting qubits and the ability to detect and counter inevitable experimental errors. This problem is complex, and one of the main avenues of research along this direction consists of error-correction techniques dedicated to locate and correct the computational errors in physical qubits in real time to preserve quantum information for longer and eventually beat the error- correcting threshold allowing to store quantum information indefinitely, thus creating a logical qubit [4, 36, 37]. Widely considered an important milestone in quantum computing along with attaining quantum supremacy, achieving even a single logical qubit has not yet been demonstrated, though the break-even point for error-correcting surface codes has been passed a few years ago [38, 39]. In turn, developing advanced control methods, techniques for detection and correction of control errors, better insulating the qubit from its environment to increase its lifetime as well as eliminating other, both fundamentally physical and equipment-generated errors lies at the heart of advancing towards this goal [36, 40]. This thesis is devoted to this small yet important part of the whole big field – in-situ quantum control over superconducting qubits, and in particular, the experimental techniques involved in it. As a plethora of excellent theses outlining the historical development of cavity QED and circuit QED fields and thoroughly introducing the theoretical models used here have been written over the past decade [41–44], I will keep the introductory part fairly brief and mostly on a qualitative level. A reader interested in the derivations of Josephson equations [45], quantization of electrical circuits [46], derivations of the theoretical framework behind light- matter interaction Hamiltonians, Jaynes-Cummings model, Hamiltonian of driven qubit in different frames and representations etc I would direct towards the cited books and theses or a list of comprehensive review articles [21, 47, 48]. The experimental introduction, on the other hand, is more specific and thorough and focused heavily on technical details necessary to implement the mentioned methods and techniques.

3 CHAPTER 1. INTRODUCTION

1.4 Brief chapter description

Chapter2 serves as a general introduction to the field of circuit QED and superconducting qubits. Written from the perspective of quantum information processing, it outlines the requirements to a physical system employed as a quantum bit, introduces the concept of an artificial quantum system and motivates the usage of superconducting circuits as artificial atoms and qubits. It then introduces the constituents of superconducting circuit QED – on-chip LC- resonators and superconducting qubits. Finally, it introduces transmon superconducting qubits specifically as they are used throughout this thesis. Flux and charge qubit controls, used to apply quantum gates and change qubit’s energy splitting in situ, are briefly discussed.

Chapter3 introduces the general requirements to the cryogenic setup suitable to perform experiments with superconducting circuits. It describes the experimental setup we use while explaining how and why it is relevant to the general requirements outlined before. Starting from a global overview of the measurement setup, the chapter then introduces it in more details together with basic measurements, in a logical way layered by increasing complexity. First I give a brief overview of the hardware and wiring we are using. I then move on to describe in details the most basic parts of the setup which are used in every experiment. In measurement terms, we start from the discussion of time-independent and relatively simple spectroscopy measurements, then move on to introducing and describing basic time-domain measurements, after which we proceed to describe some of the more advanced optimal control and calibration techniques involving full potential of the machinery present. The new hardware and software tools are introduced as they become necessary. Specific attention is paid to the technical details of pulse synchronization and triggering of the equipment, details of the pulse alignment organisation, and software virtual timing instruments and pulse generating library. Properties and implications for quantum mechanical observables by performing averaged and single shot readouts are also discussed in detail. Chapter4 contains an application of the toolbox developed in Chapter3 to a practical problem of generating high-quality random numbers. Protocols certified by violation of certain Bell-type inequalities have been proposed and realized recently [49–51], thus creating a notion of certified quantum random number generators (QRNGs). We follow a different approach to QRNG certification based on the Kochen-Specker theorem and contextual measurements [52]. The Chapter presents an experimental realization of the theoretical proposal and describes its advantages over other schemes, such as higher bit rate, lifting the requirement for the input ”seed” randomness and others. 10 GBit of the generated raw data are statistically analyzed in collaboration with our colleagues from Auckland university [53]. The results of Chapter4 are published in Phys. Rev. Lett. 119, 240501 (2017) [54]. Chapter5 extends the discussion of in situ quantum control. We address the control and state preparation errors coming from pulse distortions due to imperfect qubit control lines [42, 43, 55–58]. Considering a qubit simultaneously driven by a longitudinal and a transverse

4 1.4. BRIEF CHAPTER DESCRIPTION microwave drives under certain resonant conditions, we show that in the rotating dressed-state picture (second rotating frame) one can get information about both the amplitude and the phase of the drive signals. This allows us to employ a qubit as an on-chip vector network analyzer and precisely measure the transfer functions of its control lines. We describe the theoretical idea underlying our method and extend the theoretical description to a non-resonant driving case allowing us to extend the frequency range of our method beyond any of the existing ones in the high-frequency part of the spectrum. We introduce the experimental procedure and give directions for its efficient implementation. Following that, we test the method and apply it to a standard cQED system of two coupled transmons. Character- izing out the errors of control lines allows applying better control pulses, which we demonstrate by showing an increase in fidelity of an entangling CPHASE gate. Finally, the fundamental accuracy limits of the reconstruction technique and its error budget for the particular experimental implementation on our system are presented. The results of this chapter are published in Phys. Rev. Lett. 123, 150501 (2019) [59]. Chapter6 concludes the results of the thesis and develops a tool helpful in initialization of a qubit in a pure state, which is a prerequisite for quantum computer operation [4, 60]. More specifically, we show that the decay of the correlations between two sequential measurements of a quantum bit is a manifestation of its nonzero effective temperature. We use it to develop a precise method of evaluating the qubits population allowing for virtually unlimited suppression of absolute errors. Similar to Chapter5, we start from presenting the theoretical idea of our method on an abstract ideal quantum bit with an instant and noiseless QND measurement. We then discuss experimental enhancements allowing one to suppress the measurement noise and reach the accuracy of up to 0.01%. We repeat the study of residual excited state population of a transmon qubit vs the temperature of the mixing chamber plate of a dilution cryostat [61]. High precision of our method allows us to demonstrate a qualitatively novel result of qubit being coupled to two separate thermal baths. We introduce the ‘temperature spectroscopy’ as a tool to acquire more information about the qubits spurious excitation sources. Finally, we discuss the error contributions to our method from various sources and compare it to the error sources of two conventional methods – single shot counting and exploiting the higher transmon levels [61, 62]. The results of this chapter are published in Phys. Rev. Lett. 124 240501 (2020) [63].

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10 CHAPTER 2

ARTIFICIAL QUANTUM SYSTEMS AND SUPERCONDUCTING QUBITS

A fundamental concept of quantum information processing is a quantum bit, or a qubit – a discrete two-level quantum system. In order to realize any quantum algorithm in practice, quantum bits should be represented by some physical systems. One ought to be able to dynamically control their state with high quality of operations, protect the systems from noise and decay, and be able to perform quantum measurements [1]. Naturally, there are several concurrent approaches to realize high-quality control and measurement of physical quantum bits. This chapter gives a brief overview of the basic principles underlying the development and requirements to such platforms and introduces an important concept of an artificial quantum system. Further, we introduce the physical implementation of qubits achieved in the field of superconducting circuit QED. Superconducting circuits are currently considered to be one of the most promising technologies to achieve robust and scalable quantum computers [2,3], which is highlighted by recent successes by commercial companies – Google Sycamore chip with 53 superconducting qubits [4] used to demonstrate quantum supremacy [5] and IBM cloud quantum computing initiative [6] with multiple quantum processors available and accessible to general public and scientific community to perform experiments, with the latest processor also containing 53 qubits [7]. While these devices are yet far from the first useful commercial application, they already demonstrate the reliability and scalability of the superconducting qubits platform.

2.1 Natural qubits

As a candidate for a physical system to be employed as a qubit (or a qutrit, a quantum three- level system), let us consider a natural atom. Its characteristic energy spectrum is anharmonic, which makes it suitable: applying a coherent radiation at the frequency of S0⟩ − S1⟩ transition allows a coherent control of it (details will be explained later). Anharmonicity of the energy spectrum means that coherent driving of the S0⟩ − S1⟩ transition is oﬀ-resonant to other, higher transitions, and we do not excite higher levels of the atom and do not go out of the qubit subspace. Therefore, anharmonicity of the energy spectrum is a key requirement when one wants to employ a multi-level quantum system as a quantum bit.

11 CHAPTER 2. ARTIFICIAL QUANTUM SYSTEMS AND SUPERCONDUCTING QUBITS

An atom is one example of a natural quantum system which we can use as a quantum information processing bit. It has inherently high coherence and can preserve its state for long time; Also its simplicity leads to a relatively simple description. Historically, due to their simplicity and natural advantages, natural quantum systems such as photons and trapped ions were the ﬁrst platforms to realize quantum bits on and implement quantum algorithms [8,9]. Natural quantum systems have, however, serious Figure 2.1: An atom as a natural qubit 171 + limitations. For example, all Ytterbium ions ( Yb ), (qutrit) often being employed as qubits for currently advanced and developing platform of trapped ions [10], have exactly the same transition frequencies. Overall, many of their parameters are ﬁxed by nature and can not be tuned during the device preparation, let alone in-situ during the experiment. Most importantly, high coherence of natural quantum systems is closely related to their being well-isolated from the (dissipative) environment. While isolating a system well enough would yield the desired high coherence, it also renders achieving the strong coupling of the system in question to the control lines or other systems to be an exceptionally complicated task, thus eventually limiting the ability to control it [11].

2.2 Artiﬁcial quantum systems

These complications stimulated the development of artificial quantum systems, which, together with all their parameters, could be designed and fabricated according to the requirements of a particular experiment or task. Most importantly, careful engineering of the devices’ parameters allows to circumvent the problem of low coupling while having high coherence: the ideas behind it include isolating only specific (detrimental) parts of the spectrum [12, 13], making the device insensitive (at least in the first order) to specific types of noise while preserving the full controllability [14], the fact that the noise is usually weaker than the control signal and it is possible to fabricate a mirror which reflects noise but is transparent to the strong control signals due to saturation [15] and others. Artificial quantum systems are naturally more ’dirty’ as they are based on the solid-state architecture with inevitable defects, so the coherence times of the first superconducting qubits were very short (about 10 ns) [16]. However advances in material science and nanofabrication, better understanding of noise sources and new designs allowed to increase the coherence times to hundreds of microseconds [17], i.e. more than by four orders of magnitude while keeping the characteristic one- and two-qubit gate lengths virtually unchanged on the order of tens of nanoseconds. Concurrently with the advances in coherence times, it has been possible to

12 2.3. SUPERCONDUCTING LC CIRCUIT engineer reliable tunable strong couplings between qubits, thus ensuring the scalability of the architecture. These developments together with the recent technological advances mentioned in the Introduction better than anything else illustrate the vision of artiﬁcial quantum systems as promising building blocks for a quantum processor [2, 18].

2.3 Superconducting LC circuit

As an example of an artiﬁcial quantum system, let us consider an ideal harmonic LC-oscillator. Of course, at room temperature the lumped-element scheme shown in Fig. 2.2 is just an ideal abstraction, as each element (and the wires) has some resistance associated with it. However fabricating the device out of superconducting materials and cooling below critical temperature

Tc for those materials allows getting rid of these unwanted resistances, and the ideal lumped- element scheme becomes a valid abstraction. Recalling the energies stored in an inductor and a capacitor, we can write a (classical) Hamiltonian of an LC circuit:

q2 Φ2 H = + , (2.1) 2C 2L where q and Φ are charge stored in the capacitor and ﬂux through the inductor, and C and L – the corresponding capac- itance and inductance. Both a capacitor and an inductor are linear elements (ﬂux and voltage are directly proportional to current and charge respectively), which leads to their energies being quadratic functions of the variables. As q and Φ are canonically conjugate variables, we can follow the standard formal quantization procedure [19] and consider the quantum-mechanical description of the LC-oscillator. Enforcing the commutation relation Φˆ, qˆ = ih̵, we now consider variables q and Φ as operators Figure 2.2: An ideal LC- oscillator qˆ and Φˆ with the Hamiltonian

qˆ2 Φˆ 2 Hˆ = + . (2.2) 2C 2L Before proceeding to recall the second quantization (energy eigenstate) representation and harmonic spectrum, we would do a rescaling of variables to make the comparison with the qubit Hamiltonian shown below more evident. Introducing a magnetic ﬂux quantum [20, 21]

̵ πh −15 Φ0 = ≈ 2.068 · 10 Wb (2.3) e and redeﬁning

qˆ 2πΦˆ nˆ = ; φˆ = , (2.4) 2e Φ0 one can note the commutation relation for the ’reduced’ variables

13 CHAPTER 2. ARTIFICIAL QUANTUM SYSTEMS AND SUPERCONDUCTING QUBITS

2πΦˆ qˆ 2π [φ,ˆ nˆ] = , = ih̵ = i (2.5) Φ0 2e 2eΦ0 and write down the Hamiltonian in their terms.

1 H = 4E nˆ2 + E φˆ2. (2.6) C 2 L 2 Here EC = e ~2C is called charging energy and has a meaning of the amount of energy required to move an electron (half of a Cooper pair) from one side of the circuit to another, 2 and EL = (Φ0~2π) ~L is called inductive energy [22]. Proceeding to the second quantization to express the Hamil- tonian in terms of the energy eigenstates of the system would result in a textbook Hamiltonian of a quantum harmonic oscillator: 1 Hˆ = hω̵ aˆ†aˆ + , r 2 √ (2.7) 1 8EJ EC where ωr = √ = ̵ . LC h The above-mentioned linearity of the elements yields a harmonic equidistant spectrum, where energy eigenstates are sepa- Figure 2.3: Harmonic spec- ̵ trum of an LC-oscillator rated by the same energy hωr (Fig. 2.3). While we can not directly use an LC-oscillator as a quantum bit, it forms an eﬀective base to implement a superconducting qubit. As we have been emphasizing the linearity of the circuit causing the quadratic Hamilto- nian and the harmonic spectrum, it is intuitively evident that what is required is to add a little non-linearity, which would yield the desired anharmonicity, while preferably, of course, keeping the circuit resistless. An actual LC-resonator fabricated on-chip can be cooled down to its ground quantum state in a dilution cryostat and shown to follow its quantum description, thus embodying an example of a mesoscopic quantum system, i.e. macroscopic system comprised of a huge number of atoms and electrons, but still exhibiting the behaviour of an individual quantum system with quantized energy levels and well-deﬁned structure [11, 23]. In fact, though not used as qubits, superconducting LC-resonators are essential constituents of the circuit QED quantum computing architecture as they are used to read out the state of superconducting qubits and can be used as a quantum bus [18].

2.3.1 Superconducting qubit

The required non-linearity together with non-dissipative nature of the element is provided by a Josephson junction, which forms a fundamental building block of superconducting circuit QED [2, 11]. Josephson junction consists of two superconductors coupled by a weak link, which in case of fabricated devices is usually represented by a thin insulating layer. Discovery made in

14 2.3. SUPERCONDUCTING LC CIRCUIT

1962 by B.D. Josephson was to note that in such configuration a current will flow between the superconductors without resistance; moreover, its value depends only on the phase difference between the wavefunctions corresponding to the two superconductors [24]. This effect was named a DC Josephson effect and is described by a following relation:

Is = Ic sin ϕ, (2.8) where Is is a supercurrent flowing between the two superconductors, Ic is called critical current and represents the maximum amount of current which can flow through the junction, and ϕ is the superconducting phase difference [20, 25]. A second effect, called an AC Josephson effect, happens if one applies constant voltage V through the junction. In this case the supercurrent flowing through the junction becomes alternating with the frequency

2eV ω = . (2.9) J h̵ This eﬀect can be understood, recalling Eq. (2.8), as the constant voltage changing the superconducting phase diﬀerence:

2eV I (t) = I sin ϕ + t , (2.10) s c 0 h̵ or for the phase diﬀerence ϕ

2eV ϕ˙ = ̵ . (2.11) h Figure 2.4: A non-linear LC- From the above two relations (2.8) and (2.11) connecting the oscillator. Marked with a cross is a Josephson junction. current and voltage through the junction and the phase diﬀerence ϕ,which is also the reduced ﬂux we introduced earlier, one can calculate the energy stored in the Josephson junction.

t t ′ ̵ ̵ ′ ′ ′ ′ ϕ˙(t )h ′ Ich EJJ = S V (t )I(t )dt = S Ic sin ϕ(t ) dt = (1−cos ϕ). −∞ −∞ 2e 2e (2.12)

This energy can be expressed as EJ (1 − cos ϕ), where EJ is a constant called the Josephson energy, and has two important properties. First, its Taylor series starts from a term quadratic in ϕ, and therefore it closely resembles the energy stored in a normal inductor provided that ϕ is small. And second, it has higher orders which allows us to consider it as a non-linear inductor and employ in making an artiﬁcial quantum system. Let us now consider an LC-oscillator, but substitute the inductor by the new element – a Josephson junction (see Fig. 2.4). The Hamiltonian of the circuit would read

15 CHAPTER 2. ARTIFICIAL QUANTUM SYSTEMS AND SUPERCONDUCTING QUBITS

2 H = 4EC nˆ + EJ (1 − cos φˆ). (2.13)

Naturally, the ﬁrst-order expansion of it reproduces the

Hamiltonian of an ideal LC oscillator (2.6): EJ (1 − cos φˆ) ≈ 1 ˆ2 2 EJ φ , and the Josephson energy takes the role of inductive energy in the LC oscillator. Curiously enough, this expansion as a first approximation is not always valid. In order for us to consider the Taylor expansion of cos φˆ and treat its higher terms as perturbations, we should ensure that φˆ is a well-defined quantum number which is also close to zero. Whether it is true depends on the energy scales Figure 2.5: Anharmonic spec- of the system: if EC ≫ EJ or EC ≃ EJ , φˆ is not localised and trum of a nonlinear system exhibits strong fluctuations. Therefore, the expansion is not valid comprised of a capacitor and a Josephson junction and one has to find other ways to calculate the spectrum of the Hamiltonian (2.13).

2.4 Transmon qubit

In case of EJ ≫ EC , on the contrary, the potential due to the Hamiltonian (2.13) becomes cosine-like following the dominating term with EJ with only a small disturbance due to the

EC term, and φˆ becomes a well-deﬁned quantum number [14]. Consequently, expanding cos φˆ in Taylor series around zero is a valid mathematical treatment and one can directly calculate 1 ˆ4 the spectrum by considering the next term (− 24 EJ φ ) in the expansion. Treating it as a perturbation to the harmonic Hamiltonian yields the anharmonic level structure: √ 8E E − E ω = J C C ≡ ω , 01 h̵ q ̵ ω12 = ωq − EC ~h, ̵ ω23 = ωq − 2EC ~h, (2.14) ⋮ with ∞ i ̵ Hˆ = h Q Q ωk,k+1 Si + 1⟩⟨i + 1S , i=0 k=0 or in terms of creation and annihilation operators

E Hˆ = hω̵ aˆ†aˆ − C aˆ†aˆ†aˆa.ˆ (2.15) q 2

The frequency difference between the first two transitions of the system is called anharmonicity (denoted α). Anharmonicity also defines, to the first order, the spacing between the next transitions (not levels!) in transmon case, thus allowing one to use it as a multi-level

16 2.4. TRANSMON QUBIT quantum system (qutrit, or more levels) due to clearly resolvable frequencies.

E α = ω − ω = − C ≃ ω − ω ... (2.16) q 12 h̵ 12 23

Transmon qubits are conventionally fabricated to work in the microwave regime with transition frequencies of 4-12 GHz and anharmonicity on the order of several hundred MHz [14, 22]. Typical level structure of a transmon qubit up to the third energy level is shown in Fig. 2.6. Note the anharmonicity is negative due to the 1 ˆ4 sign of the next expansion term of cos (− 24 EJ φ ), and can not be made arbitrarily large. In terms of charging and Josephson energies EC and EJ typical values Figure 2.6: Typical energy spectrum of ̵ ̵ are EC ~h ∼ 200 − 400 MHz and EJ ~h ∼ 10 − 50 GHz. a transmon qubit. First three levels allow operating it as a qubit (only the ﬁrst Polynomial decrease of anharmonicity is the tradeoﬀ transition), or a qutrit – quantum three- for the exponential suppression of the charge noise in level system, both transitions. the transmon qubit (based on EJ ~EC ratio).

Presence of the higher levels poses certain complications and has to be taken into account, particularly for strong driving (short gate times) and interactions [26]. On the other hand, some of the qubit operations take advantage of these higher levels. For instance, a coherent exchange of an excitation between S1q11q2⟩ and S2q10q2⟩ allows realising an (entangling) CPHASE gate between two superconducting qubits, and util- ising the S2⟩ state of a single transmon with a coupled resonator enables its fast active reset [27, 28]. Those protocols will be discussed in more detail in following chapters.

Side note: Historically, the regime EJ ≤ EC was explored first by the superconducting community. While the calculations presented above are not valid for that case, one can still demonstrate their spectrum to be anharmonic; furthermore, the desired anharmonicity is much stronger in case of EJ ≤ EC . First experimental implementation of a superconducting qubit was based on this regime (called a Cooper-pair box) [16], as well as most of the early results in the field [29]. However the first qubits were relatively short-lived and suffered from many decoherence channels. In 2007 it was demonstrated that exploring the inverse regime EJ ≫ EC greatly decreases the sensitivity of qubits to offset charge noise while keeping anharmonicity large enough and retaining the ability to perform coherent manipulations with them [14]. While schematically represented by the same circuit, qualitatively different properties and greatly enhanced decoherence times earned them a separate name of transmon qubits. Virtually all qubits used in the commercial efforts to make a useful universal quantum computer are currently of various transmon sub-types (such as X-mon [30]), though other qubit modalities (such as Cooper pair box, or flux qubits) are used in research applications.

17 CHAPTER 2. ARTIFICIAL QUANTUM SYSTEMS AND SUPERCONDUCTING QUBITS

2.5 dc-SQUID and tunable transmon qubit

Let us now consider two Josephson junctions forming a loop1. As magnetic ﬂux through the loop is quantized, the phases over the junctions sum up to the external ﬂux modulo 2π [20]:

2πΦ ϕ1 − ϕ2 = . (2.17) Φ0 Assuming the junctions to be identical2, the current ﬂowing through them is

ϕ1 + ϕ2 ϕ1 − ϕ2 Is = Ic(sin ϕ1 + sin ϕ2) = 2Ic sin cos . (2.18) 2 2 Figure 2.7: SQUID loop con- Defining the average phase as ϕ = ϕ1+ϕ2 , one can see that sisting of two Josephson junc- 2 tions with external magnetic the current-phase relation for the SQUID loop is similar to the flux applied. Josephson relation for a single junction, but with the effective critical current depending on the external flux Φ:

πΦ Is = 2Ic cos sin ϕ = I˜c(Φ) sin ϕ. (2.19) Φ0 One can make use of this similarity by substituting a single junction in the qubit scheme by a SQUID loop (see Fig. 2.8). The critical current dependence leads to the eﬀective Josephson energy in the qubit Hamiltonian (2.13) being dependent on Φ:

I (Φ)h̵ E = c ≡ E (Φ), (2.20) J 2e J and so the energy splitting of the transmon qubit with a SQUID loop becomes in-situ tunable by the external magnetic flux: » 8EJ (Φ)EC − EC ω = . (2.21) q h̵ Note, that according to Eqs. (2.18), (2.19) the maximum qubit frequency is achieved when the external flux is zero, and EJ takes its maximum value. Changing the external flux decreases (and then increases periodically) the qubit’s frequency, but reaching low values leads to the breakdown of the transmon regime EJ ≫ EC due to EJ getting small enough to be comparable to EC . Finally, we can do the last step and consider this multi-level system as a qubit by restricting the operations only to the first two energy eigenstates of the transmon Hamiltonian (2.15).

1This loop of Josephson junctions bears a name of superconducting quantum interference device (SQUID, or DC SQUID) and is widely used as a sensitive magnetometer capable of measuring weak magnetic ﬁelds. Australian industry, for example, has beneﬁted from SQUID-based systems for geomagnetic exploration of minerals by discovering major mineral ore deposits. 2In case of an asymmetric SQUID the maths gets a bit less simple, but the qualitative conclusions remain

18 2.5. DC-SQUID AND TUNABLE TRANSMON QUBIT

Figure 2.8: Full scheme of a tunable transmon qubit. A single Josephson junction is replaced by a loop containing two junctions, which makes the qubit frequency in-situ tunable by the external magnetic ﬂux. The loop is shunted by a large capacitor Cg leading to a small charging energy EC . Parasitic capacitances of the junctions are shown for the completeness of the model.

Writing it in a Pauli matrix representation, the Hamiltonian of the qubit simply becomes

h̵ H = ω σ . (2.22) 2 q z As we have seen, changing magnetic ﬂux through the SQUID loop allows one to change ̵ qubit’s energy splitting hωq. However to work with the expressions it is more convenient to explicitly separate a ﬁxed time-independent qubit frequency ωq (for instance, the value at zero time) and a time-dependent changing contribution. Therefore, in case of a tunable transmon we can add an extra term responsible for this external tuning into the Hamiltonian,

h̵ H = ω σ + hA̵ (t)σ , (2.23) 2 q z z z

where Az(t) represents the time-dependent part of magnetic ﬂux. Applying a microwave drive tone coupled capacitively to the superconducting qubit leads to energy exchange between the qubit and the drive. In the reduced picture when we consider only the ﬁrst two energy eigenstates, it corresponds to coupling to the σx operator in the Hamiltonian (transverse coupling) and allows applying gates to the qubit (changing its state)

[31, 32]. More speciﬁcally, a microwave drive with frequency ωx, phase φx leads to an extra driving term in the Hamiltonian

̵ hAx cos (ωxt + φx) σx, (2.24)

Where Ax represents the driving amplitude and depends on the strength of the drive and its coupling to the qubit. Unlike the σz coupling which only changes the energy splitting of a qubit, microwaves coupled through the σx term induce transitions and therefore microwave pulses are used to realize quantum gates on the superconducting qubits.

19 BIBLIOGRAPHY

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22 CHAPTER 3

EXPERIMENTAL TOOLBOX

This chapter focuses on experimental techniques and methods we have used to successfully perform experiments with superconducting qubits and aims to give an overview of a toolbox we have employed. It is intended to be suitable as an introduction or a guide to performing experiments with superconducting circuits, so particular attention is paid to technical details non-essential from the theoretical point of view, which could, however, cost hours of work to ﬁx without prior knowledge of them.

3.1 General requirements

In order to operate systems based on superconducting circuits and perform quantum information processing or fundamental light-matter interaction experiments, one has to meet several basic criteria. Most naturally, in order to operate superconducting devices one has to ensure the operating temperature is lower than Tc for the material of choice (conventionally used aluminum and niobium have Tc of 1.2 K and 9.25 K, respectively). However even stricter limitation comes from the necessity to reach the ground quantum state of the system. One of the fundamental requirements for a quantum processor is the ability to initialize the qubits to a simple fiducial state [1], as many quantum algorithms require a known input state to effectively operate. It is conventionally achieved in a field of superconducting circuits by waiting for the qubits to thermalize with their low-temperature environment, thus cooling them down ̵ deeply into the ground state [2,3]. It means one has to ensure the condition kBT ≪ hωq, so the thermal population of the excited state of the system in equilibrium with the environment −hω̵ ~k T Pe ≈ e q B is small. Recalling the operating frequencies ωq are on the order of several (4-12) GHz and expressing it in the units of temperature (1 GHz ≈ 48 mK), we can reformulate the condition to be T ≪ 200 mK, where T is the operating temperature of the device. Superconducting circuits are therefore operated under high vacuum (to be well-isolated thermally from hot environment) inside 3He~4He dilution cryostats capable of reaching temperatures of tens of mK. Dilution refrigerators operate a mixture of 3He and 4He in various phases as a coolant. Comprised of several temperature stages for technological reasons, they are ultimately based on the fact that at zero temperature 3He and 4He do not separate fully, but split into a pure 3He phase and a dilute 3He~4He phase with finite 3He concentration of about 6.6%.

23 CHAPTER 3. EXPERIMENTAL TOOLBOX

The enthalpy difference between 3He in the dilute and 3He in the concentrated phases allows realizing a continuous closed-cycle cooling process and offers the only continuous refrigeration method for temperatures below 0.3 K1 [4]. Modern cryostats are capable of achieving 10 mK base temperature with enough cooling power to sustain the operation of 100-qubit systems with full wiring and controllability [5]. Another prerequisite is sufficient isolation from the external radiation and electromagnetic fields. In addition to a vacuum-tight outer vacuum can (OVC) shield, the cryostat itself has three extra inner shields connected to a 70 K, 4 K and Still (≃ 0.7 K) plates. The samples are mounted on and thermally anchored to the Mixing Chamber (MC) plate of the cryostat with the base temperature of ≲ 20 mK. Each sample is enclosed in two additional shields. The external one is made of µ-metal, which is effective at screening slow and constant magnetic fields across a wide range of temperatures (from room temperature (RT) to cold) and the internal one is made of aluminum and repels the rest of the permeating magnetic fields whenever the cryostat cools down below 1.2 K (Tc for aluminum). Finally, despite the extremely low temperature and well-protected environment, one has to preserve a way of coherent manipulation of the samples mounted within the cryostat. It requires multiple control DC- and microwave lines with sufficient attenuation to suppress the Johnson- Nyquist thermal noise and decrease the control and readout signals to the weak level required by quantum devices. In reverse, several output lines together with a multi-stage amplification chain are necessary to enhance the amplitude of weak readout signals to be registered by room- temperature electronics with high thermal and electronic noise. To put somewhat ambiguous word ”weak” into perspective, one photon in a 5 GHz resonator corresponds to 3.31 · 10−24 J of energy. Populating the resonator with a photon over 1 µs corresponds to 3.31 · 10−18 W of power, or about -145 dBm. This value ballparks the input and output power for a continuous spectroscopic measurements, where one has to ensure the average amount of photons in the resonator is sufficiently less than one. For pulse-based time-domain operation of a quantum processor the typical power values are 15-25 dB higher.

3.2 Experimental setup

Our experimental setup consists of three general parts. Room-temperature hardware is used to generate the pulses and control the experiments. Cryostat wiring is passive and relatively ﬁxed, yet crucial to deliver the control signals to the lowest temperature stage. The last part consists of cold wiring at the base plate of the dilution cryostat and samples themselves.

3.2.1 Room-temperature setup

Room-temperature part of the setup is the most diverse and consists of a plethora of very diﬀerent devices. Used to generate manipulation and measurement pulses, control the experiment timings, set the parameters of quantum systems, readout and process the output

1Book [4] contains an easily readable introduction to the principles of 3He~4He dilution cryostats operation

24 3.2. EXPERIMENTAL SETUP information and calibrate out possible errors, it is a driving force behind all the physics experiments and quantum computation protocols. The importance of designing and assembling the room-temperature hardware properly is hard to overstate. As we have mentioned in the previous chapter, superconducting qubits and resonators have energy splittings in the several GHz range [6,7], so one can use well-established and precise microwave electronics and equipment which has been under development for several decades and has a high degree of precision and reliability. In order to generate control pulses and drive the qubits, the first device one needs to use is a microwave (MW) source, or a signal generator. MW sources are capable of generating a stable in power and phase signal in a wide range of frequencies in the radio frequency (RF) band. For example, one of the sources we use, a Rohde & Schwarz SGS 100A RF source has a frequency range of 1 MHz – 12.75 GHz with a power range of -120 to +25 dBm [8]. Microwave sources alone might be enough to perform continuous-wave experiments (spectroscopy) or perform qubit’s readout through a coupled resonator, which typically requires several microsecond-long pulses (more details below in the chapter). Manipulating the qubits however requires much shorter pulses (8-80 ns) [9], and microwave source might not be able to switch on these timescales. In addition, one may require pulses with specific shape. For example, pulses with Gaussian envelope shape (as opposed to square envelope shape) are widely used to make the pulse spectrum confined in the frequency domain and therefore better isolate it from unwanted transitions [10, 11]. Cancelling the distortions of control lines (see Chapter5) requires custom pulse shape, which is not possible with a standard signal generator. To meet both requirements and also get the control over the pulse phase, arbitrary waveform generators (AWGs) are used in addition to MW-sources. Digital generating devices, AWGs can output any pre-programmed waveform by sampling it over a series of discrete time points with user-specified amplitude. The length of a single time point is the inverse of AWG’s sampling rate. For an Agilent N8241A AWG used in our lab [12] the sampling rate is 1.25 Gs/s, which corresponds to one time point being 800 ps long. Naturally, as outputting a time period of a signal requires at least two points (its minimum and maximum), the output frequency of an AWG is effectively limited by its Nyquist frequency specified by half of its sampling rate. To combine the two source signals – high-frequency MW-source carrier signal and low- frequency AWG ’shaping’ signal we use an IQ-mixer (see Fig. 3.1). An IQ-mixer is a 4-port passive device which accepts a stable high-power carrier signal (called local oscillator, LO) and two control microwave signals called in-phase component (I) and out-of-phase component (Q). Provided that I and Q components have the same frequency, an IQ-mixer generates two sidebands at frequencies ωLO ± ωIQ, where ωLO and ωIQ are frequencies of LO and IQ ports, respectively. Changing the relative amplitudes and phases of I and Q input components allows controlling the amplitudes of both sidebands (including suppressing one of the sidebands to zero), and also the phase of the resulting pulses. Keeping the LO power in the specified range ensures that the mixer works in a linear regime, so the amplitude of the output signals is linearly dependent on the amplitudes of I and Q components.

25 CHAPTER 3. EXPERIMENTAL TOOLBOX

Figure 3.1: High-frequency local oscillator signal coming from a stable microwave source is combined on an I-Q mixer with a low-frequency shaped pulsed signal coming from an AWG, resulting in a pulsed high-frequency signal required to drive a qubit. Each setup (microwave source + I-Q mixer + AWG) is capable of driving multiple transitions of one qubit.

In principle, AWGs with fast enough sampling rates to generate the required pulses directly exist, but their high price range makes their usage impractical, and the up-conversion approach using mixers is widely adopted in the community. While each device generally has a notion of internal time and frequency reference via an internal clock or resonating circuit, these notions might conflict for different devices. Two microwave sources set to output the same frequency ωr will in reality output some slightly different frequencies ω1 and ω2. One has to ensure that (ω1 − ω2) · Texp – the total phase shift of one signal relative to the other over the course of an experiment with length Texp – does not limit the precision of the experiment. For the same reasons, relying on an internal clock to output the control pulses at the specified time might lead to timing errors and is not generally used. Hence another important thing to keep in mind when designing the room-temperature setup is time synchronisation and need for a common frequency reference. We use a Stanford Research Systems FS725 stand-alone 10 MHz Rubidium Frequency Stan- dard [13], which generates a very stable 10 MHz signal and distributes it to all devices to give the frequency and phase reference. A digital delay generator (SRS DG 645 [14]) is used to solve the second part of the problem and trigger the devices to start output a control signal or start data acquisition. Conventionally, a trigger it is a fixed-length square pulse which is generated at a specific time, put into a dedicated input of a generating device and triggers the device to output the next programmed pulse. Time synchronisation is a complex and non-trivial task, and it is discussed in more detail in the time-domain measurements section. Of course, to record the results of a qubit’s measurements, we need to acquire the microwave signals transmitted through the device. In order to register this coming signal, we use heterodyne detection scheme. It consists of two parts: first we down-convert high-frequency signal

26 3.2. EXPERIMENTAL SETUP to an intermediate frequency (IF) using a mixer. Intermediate frequency is typically chosen in the range of 10-100 MHz [15]; we work with a fixed 25 MHz IF frequency. IF signal is then amplified using a low-noise IF amplifier, put through a low-pass filter and sent to an input of a field-programmable gate array (FPGA). Unlike the up-conversion case of qubits driving, here

ωLO and ωRF are close with a ﬁxed frequency diﬀerence of 25 MHz. It means the two sidebands are far apart: IF sideband is 25 MHz, while the high-frequency sideband is at about double the

ωRF , and can be easily discarded with a low-pass filter. Therefore, an ordinary mixer is enough for the task, and one does not need to use an IQ-mixer. We use an FPGA with a custom-written software as an acquisition device. It performs analog-to-digital conversion (ADC), digitally down-converts the IF signal to DC, applies finite impulse response (FIR) filtering and optional integration and averaging and transfers the registered data to the data processing PC. The scheme of FPGA data processing is shown in Fig. 3.2.

Figure 3.2: An FPGA accepts a frequency reference clock input, two data inputs (ADC 1 and 2), and an acquisition trigger input. Math module is optional and allows performing amplitude, power or correlation measurements between two FPGA channels. This Figure is adopted from [16]. For more details on principles of FPGA operation, programming, connections and descriptions of internal modules see Appendix A in [16].

It is not possible to register and digitise the high-frequency signal of several GHz directly due to the limited bandwidth of the ADC device. Our FPGA is capable of registering signals up to 50 MHz; A separate fast digitizer has a higher limit of 250 MHz, which is still far from resonator frequencies of 7-11 GHz. On the other hand, using a heterodyne detection scheme with an intermediate step – down-conversion to the IF frequency, registering it and performing the final down-conversion digitally – allows better sensitivity and suppresses 1/f noise compared to the direct down-conversion to DC [15]. Following Sec. 2.5, replacing a single Josephson junction with a SQUID allows in situ tuning of the qubits energy splitting by applying external magnetic field through the SQUID loop. Natural requirement to this applied magnetic field is its high stability to prevent qubit dephasing due to the jitter in its frequency. We use stable SIM 928 voltage sources with extra low-pass filtering for that purpose.

27 CHAPTER 3. EXPERIMENTAL TOOLBOX

Concluding the room-temperature part of the setup are several devices which are not necessary to operate a quantum processor, but prove invaluable to characterise the rest of the setup and to resolve any arising problems. I introduce them here for completeness. A digital serial analyzer (oscilloscope) is of great aid to set up proper time alignment of pulses; we use 4-channel Tektronix DSA 70804B. Its bandwidth and sampling rate are high enough (8 GHz and 25 Gs/s) to capture the control pulses directly in time-resolved manner, which allows to calibrate the timing delays of various instruments and between the pulses, detect any errors in hardware and software during time-resolved pulse generation and directly observe the correctness of the IQ-modulated pulse shape. A vector network analyzer (VNA) is another device useful for the setup characterization and will be introduced in Sec. 3.3.1.

3.2.2 Cryostat wiring

The second principal part of the setup is cryostat wiring. As outlined in the general requirements section 3.1, running the QIP protocols with superconducting circuits or physics experiments requires the qubits to be deeply in the ground state, which is achieved via a multi-step cooling process within the well-isolated environment of a dilution cryostat. However we also need to deliver microwave (and DC-) signals to our quantum processor and qubits, which requires extra connections between the cryostat stages and naturally introduces extra passive and active heat loads. Therefore, this cryostat wiring should a) have low thermal conductivity and not introduce high passive heat load due to a heat link between the cryostat stages; b) have sufficient attenuation to sufficiently suppress the RT-noise; c) not dissipate excessive amount of heat to stay below mixing chamber plate cooling power, which puts an upper bound on the amount of attenuation; d) be well-isolated electrically in order not to introduce electrical loops. The cryostat wiring naturally forms a link connecting room-temperature part of the setup at 300 K and high electronic and thermal noise with the well-isolated environment of the mixing chamber plate setup and quantum processor. Therefore, this wiring offers a way for thermal noise and radiation to come through and disrupt the operation of the quantum processor. Passive heat load – A dilution cryostat has several temperature stages; for an Oxford Instru- ments cryostat the stages are 70 K, 4 K, Still plate (∼0.7 K), Cold plate (∼0.1 K) and Mixing Chamber plate. As a rule of thumb, the warmer the stage is the more cooling power it can offer. Therefore, for wiring connecting the stages all the way from RT to MC plate it is important to thermalize the passing lines with every stage, thus optimising both the passive and active heat loads and allowing more incremental temperature gradient. Active heat load – Active heat load comes from applying pulses and control signals through the control lines. These signals are generated at RT with relatively high amplitude, so that the RT thermal and electronic noises are negligible compared to the signals. Required input signals on the input to the quantum processor are much lower and are below -120 dBm (Sec. 3.1), so one typically has strongly attenuated lines. Whenever a signal goes through an attenuator, most of its power transforms to heat and is dissipated at the attenuator. For example, for a

28 3.2. EXPERIMENTAL SETUP

20 dB attenuator dissipated heat is 99 times greater than the power of the transmitted signal. Therefore, attenuators are to be carefully distributed between the stages to avoid overheating the lowest temperature plates. Particular care has to be taken to thermally anchor attenuators to the corresponding state plates as their temperatures under heavy heat dissipation might differ significantly if the thermalization is poor. Noise attenuation – Power of Johnson-Nyquist thermal noise is proportional to the temperature of its source [4]; the MC plate temperature of 20 mK is more than five orders of magnitude lower than room temperature of 300 K. Therefore, attenuating the thermal noise from RT to MC temperature level requires more than 50 dB of attenuation. In practice, electronic noise of RT generating equipment is stronger than its Johnson-Nyquist noise, so more attenuation is required. We use the total of 60 to 70 dB of discrete attenuators in addition to the attenuation of the cables on the input lines. A recent paper [17] has shown an increase in qubit coherence

(T2* time, see Sec. 3.4.11) due to even stronger attenuating the input (and output) lines and the coupled resonator. Output lines – As the input lines are to have significant attenuation, the internal losses of cables do not matter as much with minimizing the passive heat load being the first pri- ority. Thus, we use cables made of a material with lower thermal conductivity at cryogenic temperatures. For the output signals however the situation is different: preserving the signal as much as possible until it passes the first amplifiers is of critical importance, as it affects the fidelity of readout and overall levels of noise. At the same time the number of output lines is much lower than the number of input lines, so the requirements for low thermal conductivity could be slightly relaxed. With that in mind, we use superconducting cables on the output lines as they have zero resistance and therefore do not attenuate the output signal. As it will be discussed in Sec. 3.2.5, the first amplifier conventionally used is the high-electron-mobility transistor (HEMT) amplifier located at the 4 K plate of a cryostat. Therefore, we use a material with higher Tc so the output cables are superconducting and lossless at least until this first amplifier. Our choice of material is niobium, which has a Tc of 9.25 K. A recent paper describing the design of wiring for a 100-qubit scale superconducting circuit systems [5] addresses all the mentioned requirements in great detail. In general, designing better cryogenic setups is an active area of research, and it is particularly important as the amount of qubits in a quantum processor, and therefore the amount of input lines, increases. Separately engineered modular approach with the wiring which can be assembled outside of the cryostat and then plugged in using clip connectors is being used by commercial companies. Another possible passage is to greatly reduce the amount of control wires coming from room temperature by placing some of the generating equipment to the 4 K stage of the dilution cryostat [18], however this research is yet in its early stages.

3.2.3 Mixing chamber plate setup

The ﬁnal part of our setup is located at the cold plate of a dilution cryostat. It consists of the samples for QIP or SQD experiments, directional couplers and circulators for signal routing,

29 CHAPTER 3. EXPERIMENTAL TOOLBOX quantum-limited amplifiers if necessary and experiment-specific wiring which connects it all. Unlike the cryostat wiring, which stays mostly unchanged, the cryo part of the setup depends a lot on the experiment we are performing and might be entirely different for a fundamental physics experiment testing nonreciprocal transmission via a waveguide with embedded qubits [19], where it is necessary to get access to both transmission and reflection measurements and also drive the device from all possible directions, and a more straightforward application of a quantum processor to generate certified random numbers (see [20] and Chapter4), where only the transmission measurements from one direction are required, but extra quantum-limited amplification via a Josephson parametric amplifier (JPA) becomes a necessity. One of the important requirements here is the presence of some nonreciprocal device both for signal routing and isolation on the output lines. Let us consider first the output of a resonator and the signal going out of it and into the HEMT amplifier. Similarly to the input lines, one has to ensure the thermal Johnson-Nyquist noise from HEMT does not propagate back and destroy the coherence of qubits coupled to the resonator. One can not use attenuators for this purpose again, however: as attenuators are reciprocal devices, they will also greatly decrease the amplitude of the output signal and render it impossible to perform high-fidelity measurements of qubits states, particularly in a single-shot regime. To resolve this problem, circulators are used as directional isolators on the output lines (see Fig. 3.3). A circulator is a 3-port passive nonreciprocal device designed to route the incoming signals between the ports. A signal arriving to the port 1 of an ideal circulator comes out of the port 2, port 2 – to port 3 and port 3 – to port 1. A real circulator can be characterized by its insertion loss and isolation. Insertion loss characterizes the loss of signal power in dB in the transmission direction, and isolation reflects the loss of signal power in transmission in the ”forbidden” direction – for example, from port 1 to port 3. A Raditek cryo circulator for the range of 4-8 GHz offers Figure 3.3: The 18 dB of insulation with 0.4 dB insertion loss [21]. Terminating the port output signal is routed to a HEMT 3 of a circulator with an impedance-matched resistor allows using it as a amplifier via a series good isolator of back-propagating noise from HEMT amplifier while not of circulators to stop attenuating the signal much. We use pairs of circulators for that purpose, the thermal noise from propagating as 18 dB is not enough to decrease the 4 K thermal noise to the 20 mK back. level. While using commercial circulators similar to the mentioned one does the job for the small-scale systems, it is not ideal and gets increasingly problematic as amount of qubits on a quantum processor scales up. Nonreciprocity in passive circulators is based on strong magnetic field, which is disruptive to superconductivity and adds spurious magnetic flux and extra noise to the SQUID loops of tunable qubits. For that reason, circulators cannot

30 3.2. EXPERIMENTAL SETUP

Figure 3.4: Output signal passes a circulator for isolation purposes and then routed to a directional coupler (DC), where a strong pump tone to activate the Josephson parametric amplifier (JPA) is added. The signal then continues through to the JPA, where it is amplified and reflected. The amplified signal is routed upwards to another circulator for isolation and continues towards an output line and a HEMT amplifier.

be realized on-chip together with transmission line resonators and qubits, but are present as separate bulky elements of cryogenic setup, thus crowding the limited space and requiring extra insulation. Realizing active or passive circulators which can be fabricated on-chip alongside superconducting circuits and do not require magnetic ﬁeld to operate is a very active research direction. Despite several recent results and theoretical proposals (see [22] and references therein), it is still work in progress, and external magnetic circulators are being used throughout the community.

An optional yet indispensable for certain applications part of a cryogenic setup is a Josephson parametric amplifier (JPA). JPAs used in SQDLab are capable of providing some 20 to 30 dB of quantum-limited amplification with bandwidth of several MHz and consist of pairs of resonators based on arrays of Josephson junctions. Our JPAs work on reflection, i.e. amplify the reflected signal. Utilizing a 4-wave mixing process, a JPA requires a strong pump tone and a signal routing to isolate the amplified reflected signal from the input signal. A circulator is used here directly as a three-port device to route the signal to a Josephson parametric amplifier and then to the output, as shown in Figure 3.4. Figure 3.5 contains an example of a whole experimental setup, including the full wiring of the mixing chamber plate part.

Finally, cryogenic switches allow alternating between the samples within the cryostat without the necessity to warm it up to the room temperature and replace the wiring. A switch requires considerable amount of current and heat dissipated to switch its output (or input), and increases the temperature of the mixing chamber plate by several tens of mK. However in the stable routing mode it dissipates no extra heat. The switches are installed on the output lines as those lines pose a limitation: they require 4K HEMT ampliﬁers having high heat load and price.

31 CHAPTER 3. EXPERIMENTAL TOOLBOX

3.2.4 Qubit controls

While from the quantum-information theory point of view in order to realize an algorithm or protocol it might suffice to be able to generate any rotation of a qubit on the Bloch sphere, which experimentally can be realized by applying a coherent microwave pulse at the qubit resonant frequency (charge driving), coupling to the σz term in the qubit’s Hamiltonian (longitudinal ’flux’ driving) proves useful for a whole variety of applications. First and foremost, it grants the ability to change the qubit transition frequency in-situ during an experiment. Avoiding spurious coupled two-level systems and defects, effectively changing the qubit-resonator coupling, realizing two-qubit gates between qubits by tuning them in and out of the resonance, modulating decay and decoherence times T1 and T2 are just a few things such in-situ control allows one to achieve. Frequency tuning also allows demonstrating the qubit-resonator coupling directly by a spectroscopic measurement. It is done by observing the coupled resonator’s frequency shift depending on the qubit transition frequency and even the direct observation of an anticrossing between the qubit and the resonator. First demonstrated in 2004 [23], it was a major step forward for the field and indicated the presence of strong coupling between the two, eventually paving the way to realizing the dispersive readout [24] and superconducting quantum computing. As demonstrated at the end of Chapter2, the Hamiltonian of a qubit with longitudinal (frequency control, ’flux line’) and transverse (excitation, ’charge line’) driven control lines could be written in the following way:

h̵ H = ω σ + hA̵ cos (ω t + φ ) σ 2 q z x x x x ̵ + hAz cos (ωzt + φz) σz. (3.1)

Note, that while most of the time we are going to send a DC-signal (constant offset) via the longitudinally coupled line, it in principle allows sending AC-pulses. We are going to utilize this possibility to calibrate the control lines later in Chapter5. Longitudinal coupling is realized by changing magnetic flux through the qubit’s SQUID loop (see Sec. 2.5). In 2D design it might be realized by applying an overall magnetic field to the chip, for instance via a magnetic flux coils attached to a sample holder. Another option is to use a dedicated flux control line consisting of a wire going close to the qubit’s SQUID loop. In 3D design, however, only the first option is available, which contributes to a poorer scaling of this type of architecture. Figure 3.5 corresponds to an experiment performed on 3D design, so the latter option is realized. While attaching a coil to the sample holder might be an easier way as it is less demanding from the fabrication point of view, individually controlling multiple qubits quickly gets complicated. Tuning the frequencies of n qubits with n coils requires inverting a nxn matrix of coil-qubits couplings (n2 calibration measurements) and finding its eigenvectors (changing the flux along those vectors would keep one of the qubits at the same flux point).

32 3.2. EXPERIMENTAL SETUP

Figure 3.5: Full simplified diagram of a measurement setup used in Chapter4. A transmon type multi-level quantum system is incorporated into a 3D microwave copper cavity attached to the cold stage of a dilution cryostat. A magnetically tunable Josephson junction loop (SQUID, see Sec. 2.5) is used to control the transition frequency of the qutrit by a superconducting coil attached to the cavity. Amplitude-controlled and phase-controlled microwave pulses are applied to the input port of the cavity by a quadrature IF (IQ) mixer driven by a local oscillator (LO) and sideband modulated by an arbitrary waveform generator (AWG). The measurement signals transmitted through the cavity are amplified by quantum Josephson parmateric amplifier (JPA), by a high-electron-mobility transistor (HEMT) amplifier at 4 K and a chain of room temperature (RT) amplifiers. JPA is also magnetically tunable allowing to match its amplification range with the cavity output (readout) frequency. The sample at 20 mK is isolated from the higher temperature stages by a series of circulators. The amplified transmission signal is down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated LO, and is digitized by an analog-to-digital converter (ADC) for data analysis.

33 CHAPTER 3. EXPERIMENTAL TOOLBOX

3.2.5 Output signal ampliﬁcation chain

Microwave pulses (or continuous driving) we use to manipulate qubits’ states only have to go ’one way’ into the cryostat. Resonator readout pulses, on the contrary, have to go all the way through and return back to the room-temperature (RT) electronics for them to be down- converted, digitized on FPGA and analyzed to infer the state of the qubits. As those signals are very weak (see Sec. 3.1), one needs to implement a multi-step amplification chain while abiding by a few restrictions. The first wide-band amplifier conventionally used is a high-electron-mobility transistor (HEMT) amplifier [5, 25], which is usually placed at the 4 K stage of a dilution cryostat. While generally placing the first amplifier in the chain to lower temperature stage would decrease the Johnson (thermal) [4] noise associated with it and therefore increase the final signal-to-noise ratio (SNR), two main factors contribute equally to this placement. First, the 4 K plate is the coldest one having high enough cooling power to sustain its heat output and not compromise the temperature of the mixing chamber. And most importantly, the noise temperature of internal electronic noise of a transistor-based HEMT amplifier is comparable to (and might even exceed) 4 K [25] making the lower placement unnecessary. A set of sequential room-temperature amplifiers is used to increase the signal further and get it into the range suitable for FPGA. This set includes one or two broadband high-frequency amplifiers and a final IF amplifier(s) after the down-conversion to intermediate frequency. A low-pass filter is added to cut the extra high-frequency noise on the input of FPGA. Since the first conventional amplifier in the chain is capable of providing above 30 dB of signal enhancement, it is its thermal noise which limits the signal-to-noise ratio of the whole amplification chain. Furthermore, the 4 K placement of a HEMT amplifier implies an important outcome: the signal gets mixed with 4 K thermal and electronic noise before amplification, thus vastly limiting the attainable SNR even if all the subsequent amplification chain is perfect and noiseless.

3.2.6 Josephson parametric ampliﬁer

The 4K thermal noise from HEMT ampliﬁer may be commensurate or even exceed the readout signal containing the information about the qubit state. It is a Johnson-Nyquist electronic noise well approximated by a white noise model, so preparing and repeating the same experiment multiple times (N) allows decreasing the resulting noise amplitude ε (with the standard scaling ε ∝ √1 of N ) while keeping the signal constant, thus actually allowing one to get access to the measurement outcomes. This technique proves suﬃcient to perform spectroscopic measurements and demonstrate qubit-cavity coupling and anticrossing between the cavity and qubit spectrum lines, show the coherent dynamics of a qubit in the form of Rabi and Ramsey oscillations, determine the decay and decoherence times and even perform a plethora of quantum algorithms. Yet another class of experiments and protocols becomes attainable once the measurement

34 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS signal is sufficiently above the experimental noise floor, therefore allowing realization of the Single-shot readout (see Sec. 3.4.6). A major step forward was achieved about a decade ago with the revival of Josephson junction based parametric amplifiers [26, 27]. Josephson parametric amplifiers (JPAs) are capable of providing more than ten-fold signal amplification while being quantum-limited, therefore putting the SNR in the domain where we can actually see the readout in the single shot and observe purely quantum effects such as quantum jumps [28]. Travelling-wave parametric amplifiers (TWPAs) have been introduced recently [29, 30]. Still being quantum-limited, they have similar amplification gains while operating in a wide frequency range, thus combining the assets of broadband HEMT amplifiers and JPAs.

3.3 Spectroscopy & calibration measurements

3.3.1 VNA

A vector network analyzer (VNA) is a commercial device used to perform continuous-wave spectroscopy, i.e. measure the transmission (reflection) of microwave signals through a line. Tailored specifically to perform spectroscopic measurements, it has some advantages in the measurement speed over our FPGA-based measurement hardware and software. Streamlined data processing and virtually no time delay due to continuously switching the probe frequency allow its measurement duty cycle to be close to 100%, yielding the higher precision at a shorter time. Digital band-pass filters allow to reduce the signal noise further without a requirement for more averaging. While for relatively short measurements, such as a single resonator spectroscopy, these advantages do not make a big difference, using VNA might decrease the measurement times significantly when performing larger sweeps (cavity spectroscopy vs qubit flux) or calibrating the gain of a parametric amplifier. We use 4-channel Keysight Agilent N5232A PNA-L Series Network Analyzer with the frequency range of 300 kHz to 20 GHz. It is integrated into our measurement software via a custom-written Python module based on the qcodes data acquisition framework [31]. It allows using the VNA inside an IPython Jupyter notebook as any other piece of equipment. Data transfer and control are via the standard LAN connection and the protocol is based on pyvisa library.

3.3.2 Resonator spectroscopy

Dispersive readout of qubit’s state is realized by observing the state-dependent change of a resonator frequency [24]. Therefore, one has to determine this resonant frequency ﬁrst in order to manipulate and readout the qubits, and the starting step in the setup calibration is a measurement of the transmission spectrum through the resonator – resonator spectroscopy. A (continuous) probe tone is sent to the input port of the resonator. If the probe tone is oﬀ

35 CHAPTER 3. EXPERIMENTAL TOOLBOX the resonant frequency, then the signal is almost entirely reflected and we observe zero to little transmission through the resonator. The resonant tone however enters the cavity and populates it with photons, with some of the photons leaking through the output port. It generates the transmission output signal, which we then route to a JPA (if present in the setup), HEMT amplifier and so on through the RT amplification and down-conversion chain and to the input of an FPGA. The resonator by design is undercoupled to its input port and overcoupled to its output port. In other words, energy loss per oscillation cycle through the in- Resonator spectroscopy put port is less than the internal energy loss per cycle; data 0.20 fit on the contrary, energy loss per cycle through the output port is greater than both losses (Qinp > Qint > Qout). 0.15 The primary reason for this is maximizing the output signal: most of the energy is lost to the output line, 0.10 thus ensuring that most of the internal photons are con- Transmitted V, a.u. verted to the output signal. Also in 3D design it brings 0.05 the opportunity to adjust the output port coupling and 0.00 overall Q factor without fabricating a new resonator; in 7.630 7.633 7.636 7.639 7.642 7.645 Probe frequency, GHz 2D design both couplings are typically fixed. Transmission through a cavity as a function of Figure 3.6: Resonator spectroscopy measurement fit to a positive Lorentzian frequency exhibits a Lorentzian shape determined by shape. Absolute value of transmitted Eq. (3.2) and shown in Fig. 3.6[24, 32] voltage is plotted on the y-axis.

(κ~2)2 S ( )S2 = S S2 S ω Smax 2 2 , (3.2) (ω − ωr) + (κ~2) where Smax is the maximum of transmission, or the maximum of the transmitted voltage. Parameter κ here corresponds to the cavity decay rate. More formally, it is the ratio of power leaking from the resonator to the energy stored: W ~E = κ. As energy stored in a resonator −κt (and amount of photons) decay exponentially as E(t) = E0e , 1~κ has a physical meaning of time over which the amount of photons in the resonator decreases by a factor of e. We deﬁne the full resonator quality factor Q as the ratio of energy stored in the resonator to the energy loss per radian of oscillation Q = E . Note the possible factor of 2π here, as W · 1~ωr in some literature Q is related to energy loss per cycle of oscillation. We follow the notation of [6, 24]. Writing the deﬁnition as is, we obtain

E ω Q = = r (3.3) W · 1~ωr κ Evidently, the transmission peak in Fig. 3.6 shows the resonant frequency, and we can obtain κ (and therefore Q) as the width of the peak at half the transmission power. Here and in the next measurements we extract the parameters from the ﬁt as it yields higher precision. Resonator spectroscopy is a relatively stable measurement in terms of input power. Unless

36 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS

Resonator vs power

1.0 7.645

7.640 0.8

7.635 0.6

7.630

0.4 7.625

0.2

Probe frequency,7 GHz .620

7.615 0.0

7.610 70 60 50 40 30 20 10 − − − − − − − Probe tone power, dBm

Figure 3.7: Resonator spectroscopy as a function of the probe tone source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. At sufficiently high power (≈ −35 dBm at the source) the resonator peak start getting distorted and is sharply replaced by a bright mode peak at around -12 dBm. we populate the cavity on average with hundreds of photons to switch its regime, the spectroscopy will yield the same frequency, and the input power in only going to affect the output signal amplitude. Applying excessive power revives the bare cavity frequency in spectroscopy, instead of the resonator peak offset by the qubit coupling, as we are interested in. Resonator spectroscopy as a function of power is shown in Fig. 3.7. A narrow cavity peak emerges from the noise floor, slightly broadens with higher power, gets distorted and eventually disappears while the bare cavity peak emerges. This is called a bright cavity regime when no quantum interaction is visible. As it is evident from Fig. 3.7, one does not have to be extra careful choosing the probe tone power for the resonator spectroscopy, as a wide range of 30 dB leads to the same results. However as a rule of thumb for all the spectroscopic measurements here and below one would like to have on average (much) less than one photon in the cavity. This requirement leads to a trade-off between the visibility of the signal and the measurement precision; we are going to discuss it later in more detail. As it has been discussed in the Room-temperature setup section, FPGA accepts a constant- frequency signal of 25 MHz. Therefore, the resonator spectroscopy requires two drive tones

37 CHAPTER 3. EXPERIMENTAL TOOLBOX whose frequencies are to be swept simultaneously with a 25 MHz offset. Probe tone is sent to the cryostat and transmitted through the resonator, and a down-conversion tone is used as a local oscillator mixer input to down-convert the signal to the intermediate frequency of 25 MHz. In uqtools measurement software [33] the Parameter class allows creating linked variables with defined set and get functions (frequencies of two microwave sources) and sweeping them simultaneously. We use it to perform the resonator spectroscopy and set the defined linked parameter to the fitted peak to use in subsequent measurements.

3.3.3 Qubit spectroscopy

Similar to the resonator spectroscopy, qubit spectroscopy is a basic calibration measurement aimed to determine the frequency of the qubit. It can also be performed in continuous wave manner and requires three microwave sources: cavity source, down-conversion source and a qubit probe source, whose frequency is to be swept. The cavity probe tone and the down- conversion source are ﬁxed at ωr and ωr - 25 MHz.

Qubit spectroscopy, -50dBm Qubit spectroscopy, -40dBm

data 1.0 data 1.0 ﬁt

0.9 0.9

0.8 0.8

0.7 0.7

0.6 Transmitted V Transmitted V 0.6

0.5

0.5 0.4

5.35 5.36 5.37 5.38 5.39 5.35 5.36 5.37 5.38 5.39 Qubit probe frequency, GHz Qubit probe frequency, GHz

(a) Qubit spectroscopy (b) Qubit spectroscopy with high power

Figure 3.8: Qubit spectroscopy measurements. Absolute value of transmitted voltage is plotted on the y-axis. (a) Qubit spectroscopy measurement ﬁt to a negative Lorentzian shape. Low-power regime ensures a single narrow dip (b) High applied power showcases the dispersive shift of the qubit frequency, thus resulting in two dips corresponding to zero and one photon in the resonator.

At this time we expect a dip in transmission rather than a peak. Indeed, if the qubit probe tone is oﬀ-resonant with the qubit, the system properties are unchanged and the ﬁxed-frequency

ωr probe tone is transmitted well through the cavity. However if we hit the resonance with the qubit and start exciting it, the qubit-resonator coupling would shift the cavity resonance frequency by χ01 and ωr would not be on cavity resonance anymore, thus leading to the decreased transmission. Figure 3.8a demonstrates a spectroscopic measurement of a qubit. Unlike resonator spectroscopy, qubit spectroscopy requires more careful choice of probe tone powers. Applying too high resonator or qubit probe tone power leads to non-negligible

38 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS population of the resonator, which in turn means that considerable amount of time there is one (or two, or more) photons in the resonator. Since the coupling between the resonator and the qubit is mutual, it causes a dispersive shift of the qubit frequency – i.e. we may observe diﬀerent qubit frequencies depending on the resonator photon population. Fig. 3.8b shows two dips corresponding to zero and one photons in the resonator. Both dips can be seen on the same spectroscopic measurement, as it is a result of a continuous-wave process with high amount of repetitions. Amount of photons in the cavity in each repetition is determined by Poisson distribution with an average depending on the strength of the resonator probe tone. Therefore, with the same measurement we can observe several dips. With higher qubit tone power, the dips are wider and one may not be able to resolve them anymore, therefore increasing the resonator probe power leads to an apparent shift of the qubit line (Fig. 3.9). The next two ﬁgures 3.10 and 3.11 show spectroscopy of a qubit as a function of resonator probe tone power and as a function of the qubit probe tone power. Qubit spec, -45 & -40 dBm Note how low resonator power is critical to the correct 1.0 data observation of the qubit. Even being far from the non- 0.9 linear regime of the resonator seen in Fig. 3.7, high res- 0.8 onator power nevertheless populates it with many pho- 0.7 tons and therefore shifts the apparent qubit frequency. 0.6

0.5 Appearance of the next dip on the spectroscopy as a Transmitted V function of qubit power (Fig. 3.11) will be discussed in 0.4 detail in the next subsection. 0.3 0.2 It might be efficient to perform the spectroscopy as 5.35 5.36 5.37 5.38 Qubit probe frequency, GHz a multi-step process. Applying high powers to both the qubit and the resonator leads to a quick measure- Figure 3.9: Qubit spectroscopy measurement with high signal visibility and allows to roughly ment with too high resonator power. estimate the frequency of the qubit transition. Consec- utive measurements with lower powers allow getting in the low-photon regime and avoid the dispersive shift of the qubit, while also narrowing its peak and thus obtaining the correct frequency as an output of the fit. An extra complication with choosing a correct qubit spectroscopy power might arise for a 3D design. In this case it is impossible to drive the qubit directly through a dedicated charge line, and one has to drive the qubit through the microwave cavity resonator. Due to the natural qubit-cavity coupling’s dependence on the qubit frequency, changing the flux makes the effective driving strength fall off strongly when detuning the qubit further with the same drive tone power.

3.3.4 Spectroscopy of the higher transitions

As we have discussed in the previous chapter, transmon is actually not a qubit, but an anharmonic multi-level system. Its anharmonicity is negative and on the order of a few hundred

39 CHAPTER 3. EXPERIMENTAL TOOLBOX

Qubit spec vs resonator power 1.1

5.375 1.0

5.370 0.9

5.365 0.8

5.360 0.7

Qubit signal5 frequency,. GHz 355 0.6

5.350 0.5 80 70 60 50 40 30 − − − − − − Resonator tone power, dBm

Figure 3.10: Qubit spectroscopy as a function of the resonator source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. The qubit dip broadens and shifts to lower frequency as resonator is populated with more than one photon on average at ≈ −45 dBm power at the source.

MHz, which is enough to resolve the transitions and address them separately, but it also leaves the opportunity to address the higher levels [6,7]. To remind the notation, the ground and excited states of the qubit are called Sg⟩ and Se⟩, the next – second excited state – is conventionally called Sf⟩. Many protocols require use of qubits or qutrits (three-level quantum systems); there has not been many proposals to encode quantum information in more levels of a system (qudit), though this possibility remains open for future research. A direct way to observe the Se⟩-Sf⟩ transition would require applying two separate qubit drives. One drive is ﬁxed at the Sg⟩-Se⟩ transition frequency determined via qubit spectroscopy, and the second one is being swept around the expected Se⟩-Sf⟩ transition. Again, when the Se⟩- Sf⟩ drive gets resonant with the Se⟩-Sf⟩ transition, we transfer some of the qubit’s population into the Sf⟩ state, thus changing the cavity signal. It is important to note that we can not predict in general case whether we will see a peak or a dip in transmission, as the cavity dispersive shift due to Sf⟩ state χ12 changes the sign depending on the detuning from the resonator [34], and can be zero identically. Alternative way requires only a single source, but a higher applied power. Qubit spectroscopy vs power in a wide range is shown in Fig. 3.11. The single qubit peak gets wider as

40 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS

Figure 3.11: Qubit spectroscopy as a function of the qubit probe source power. The signal is normalized at each vertical slice to the maximum of transmission in that slice. As the qubit power increases, the qubit dip get wider and the second dip corresponding to one photon at the resonator appears (compare to Fig. 3.8b). With even higher power two-photon Sg⟩-Sf⟩ transitions are visible.

the drive power increases; also as has been discussed, the second (dispersively shifted) qubit dip appears. With suﬃcient power, a two-photon Sg⟩-Sf⟩ transition peak emerges, allowing us to determine the frequency of Se⟩-Sf⟩ transition. As the frequency of the two-photon peak is an arithmetic mean of the two transition frequencies, knowing Sg⟩-Se⟩ allows us to learn the frequency of the Se⟩-Sf⟩ transition.

Knowledge of both ωge and ωef allows determining the anharmonicity of the qubit α =

ωef − ωge. And since the anharmonicity of the qubit to the ﬁrst order equals to EC , we can determine both the Josephson and charging energies of the qubit (EJ and EC ) using Eq. (2.14).

Note, while EC is constant, EJ changes depending on the applied external ﬂux in case of ﬂux-tunable transmon (Eq. (2.20)).

And of course, we may want to use qubit directly as a three-level system, a qutrit, and then the knowledge of the frequency of the Se⟩-Sf⟩ transition is just required.

41 CHAPTER 3. EXPERIMENTAL TOOLBOX

3.3.5 Dispersive shift

It is possible to observe the dispersive shift of the cavity directly by driving the qubit on resonance while performing cavity spectroscopy. Continuously driving the qubit puts it into a 1 mixed state close to 2 (Sg⟩⟨gS + Se⟩⟨eS). Therefore, some of the times we’re driving the cavity we’ll see the measurement of the cavity corresponding to the ground state of the qubit (thus reproducing the Fig. 3.6), and the other times we will see the measurement of the cavity corresponding to the excited state of the qubit – i.e., with the dispersive shift. Since spectroscopy is a measurement with a lot of averages, eﬀectively we observe two pictures at the same time. The cavity showcases both of its Lorentzian peaks on the very same picture, therefore embodying a great direct demonstration of the dispersive shift and our readout technique (see Fig. 3.12a).

Resonator spectroscopy Resonator spectroscopy 0.05 0.04 data data g 0.04 | i 0.03

0.03 e | i f 0.02 | i 0.02

Transmitted V,0 a.u. .01 Transmitted V, a.u. 0.01

0.00 0.00 7.625 7.630 7.635 7.640 7.645 7.620 7.625 7.630 7.635 7.640 7.645 Probe frequency, GHz Probe frequency, GHz

(a) Resonator dispersive shift (b) Resonator dispersive shifts

Figure 3.12: Resonator spectroscopy with a driven qubit transition(s). (a) Qubit’s Sg⟩-Se⟩ transition is driven, thus we observe two peaks. (b) Ax extra microwave source is used to continuously drive an Se⟩-Sf⟩ transition as well. The three resonator peaks are labelled with respect to the corresponding qubit state.

Beside being a direct way to determine the magnitude of dispersive shift, Fig. 3.12b is a nice demonstration of the idea behind the dispersive readout, the interaction term in the Jaynes-Cummings Hamiltonian and the basic physics of circuit QED. Similar to the previous subsection, observation of resonator peaks corresponding to the next states of the transmon is also possible. Curiously, this picture does not hold under strong qubit driving. If the qubit drive tone exceeds a certain threshold, the resonator cannot resolve two peaks anymore, and two peaks are replaced by a single peak in the middle [35]. This is a neat result showing how strong driving can suppress backaction of the detector on the quantum system. The theory behind it together with an experimental demonstration can be found in [35]. One useful application of this technique with strong driving can be protecting a qubit from decay and decoherence induced by reading out the resonator [36].

42 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS

3.3.6 Resonator spectroscopy vs ﬂux

(a) Resonator frequency modulation (b) Avoided crossing

Figure 3.13: Measuring resonator spectroscopy as a function of the voltage applied to the qubit bias coils. (a) Modulation of the resonator frequency due to the qubit-resonator coupling. (b) An avoided crossing between the resonator and the qubit. One can directly determine the qubit-resonator coupling g by observing this plot.

When working with a flux-tunable transmon, it might be useful to perform resonator spectroscopy as a function of flux. Due to the qubit-resonator coupling, changing the energy splitting of the qubit also changes the resonator frequency. Therefore, one can observe indirectly the presence of the qubit without driving it directly and just via observing the modulation of the resonator frequency with magnetic flux. Fig. 3.13a shows this periodic modulation of resonator frequency as a function of bias voltage applied. As seen from the expression for qubit frequency, ¼ Φ 8EJ S cos π Φ SEC − EC ω = 0 , (3.4) q h̵ where one period corresponds to applying one flux quantum through the SQUID loop of a superconducting qubit. With perfect magnetic shielding, no flux applied to the bias coil should correspond to the symmetry point of the qubit – maximum resonance frequency (3.4). However one can see in Fig. 3.13a, the symmetry point is slightly shifted from zero flux. This indicates imperfect isolation of our sample from the external magnetic fields. Apart from showing a signature of qubit coupled to the cavity, performing the sweep vs external flux allows estimating qubit-cavity coupling. It is the most evident in Fig. 3.13b, where one can observe an avoided crossing between the cavity and qubit transitions. As the transition levels hybridise and repel with the strength of the coupling term g, observing a single vertical slice of the 2D sweep picture in the avoided crossing region allows one to obtain the value of g directly by measuring the distance between two peaks on the slice. Knowledge of the strength of the qubit-cavity interaction is useful as it takes part in qubit-cavity phenomena such as Purcell effect and vacuum Rabi oscillations, but also allows calculating various other properties of qubit-cavity system such as the magnitude of the dispersive shift [7].

43 CHAPTER 3. EXPERIMENTAL TOOLBOX

Historically, the avoided crossing has been the ﬁrst direct indication of the strong coupling between the qubit and cavity demonstrated in 2004 [23]. This demonstration together with the proposal of an architecture for quantum computation based on cavity QED [24] paved the way for the development of superconducting electrical circuits as a leading platform in QIP.

3.3.7 Qubit spectroscopy vs ﬂux

Performing the qubit spectroscopy at different voltages on the magnetic coil DC source concludes basic spectroscopic calibration measurements. It allows obtaining the qubit’s energy splitting as a function of the applied voltage directly, and actually includes cavity spectroscopy vs flux. Changing the flux bias modulates qubit’s energy splitting, which in turn shifts the cavity resonant frequency, therefore one has to recalibrate the cavity first before performing qubit spectroscopy at a new flux point.

Performing a full spectroscopy vs flux is useful to precisely obtain the parameters EJ and EC , as well as the zero flux bias Φ0 and voltage-to-flux conversion constant CΦ in Φ = Φ0 + CΦVext. Fitting to a full curve allows more exact calculation of the qubit Josephson and charging energies as EC obtained from the anharmonicity is only correct to the first order, and anharmonicity itself may be subject to the spectroscopic errors coming from higher applied drive powers, as discussed in the qubit spectroscopy section.

Knowledge of Φ0 and CΦ enables biasing the qubit to a desired frequency point; If one has several qubits coupled to a matching number of magnetic coils, performing a qubit spectroscopy vs flux for each qubit-coil pair yields Φ0 for each qubit and CΦ for each qubit-coil pair. There parameters constitute the flux matrix used to bias multiple qubits to a list of required frequencies analytically. An example of a qubit spectroscopy vs flux measurement for two qubits coupled to the same coil is shown in Fig. 3.14a. An important property of a flux-tunable qubit which has not yet been mentioned is its coupling to two-level fluctuators (or two-level systems, TLS) at some frequency points. It typically leads to decrease in qubit decay (T1) time; in case of stronger coupling it may cause a hybridisation of energy levels with those of a TLS and a visible avoided crossing on the spectroscopy vs flux picture [37, 38]. These frequencies with TLS couplings are to be avoided if one aims to perform quantum information processing with the qubits. To further see the effects of TLS one can follow-up the qubit spectroscopy vs flux measurement by also acquiring T1 and

T2 times, which will be discussed in Sec. 3.4.11. High amount of spurious avoided crossings on the qubit spectroscopy vs flux picture can be an indication of errors during the fabrication process of the quantum chip. Uqtools measurement software [33] allows performing nested sweeps. As for qubit spectroscopy vs flux one needs to recalibrate both the resonator (resonator spectroscopy) and the qubit, the outer loop is the voltage applied to the flux coils, and each voltage point contains two consecutive sweeps: resonator spectroscopy followed by the qubit spectroscopy. As for some flux points we can observe an anticrossing with a two-level system, which would lead to a distorted qubit spectroscopy picture and possibly failed fit, one may use exception handlers

44 3.3. SPECTROSCOPY & CALIBRATION MEASUREMENTS

(a) (b)

Figure 3.14: Qubit spectroscopies at different voltages on the magnetic coil DC source. Each vertical slice corresponds to a single qubit spectroscopy normalized to a maximum transmission, so only qubit dip(s) are visible. (a) Two qubits’ spectroscopies are performed in the same measurement, demonstrating a different qubits’ coupling to the coil DC source. Note qubit 1 is periodic in flux having maxima corresponding to 0, 1 or two flux quanta through the SQUID loop (b) A demonstration of a spectroscopy with a sliding window for a single qubit.

(try/except loop in Python) in order to move on to a next voltage point and avoid breaking the entire measurement. Finally, instead of performing the qubit spectroscopy in a wide range of frequencies for every flux point and doing a full 2D square picture, one can automatically adjust the qubit spectroscopy frequency window either from the approximate values of EJ and EC or by sliding the window from the previous flux point (see Fig. 3.14b). Narrowing the window around the expected qubit frequency may give more than an order of magnitude speed advantage, greatly paying off the efforts of extra coding. Overall, using object-oriented features of the programming language (Python in our case) such as dynamical Parameter range being adjusted based on the result of the previous measurement instead of a static array makes the code more elegant and readable, and also increases its efficiency.

45 CHAPTER 3. EXPERIMENTAL TOOLBOX

3.4 Time-domain measurements

While spectroscopic calibration measurements could be performed in continuous-wave fashion with no requirements for timing and synchronization, full operation of a quantum processor requires applying discrete control and readout pulses properly aligned in time. It leads to an entirely new set of technical challenges and more complicated machinery. The first and the most important additional technical requirement for the time-domain measurements is the ability to control and shift the pulses in time with high precision. We typically use qubit manipulation pulses of 8 to 40 nanoseconds in length with the separation of a few (3-5) nanoseconds; Consequently, the instrument outputs have to be aligned with the precision of down to a few nanoseconds to avoid pulse overlapping and unnecessary delays. The common way to achieve such a synchronization of signals (and signal phases) in microwave engineering domain is by virtue of external trigger and external clock inputs. As consistency of the signal output of arbitrary wave generators after receiving an external trigger is of utmost importance across many application fields of microwave engineering, present devices have trigger jitters on the order of a hundred of picoseconds and below. Two types of triggering pulses we will be more specifically mentioning further, ’triggers’ and ’markers’, are only nominally different: marker can be considered as a continuous trigger, with the required event happening while the marker is on (for example, a microwave source is outputting its signal). Clock sync via external clock input serves a purpose of phase synchronization across multiple microwave sources and AWGs which may prove useful when generating various rotations on the Bloch sphere. As the phase of the qubit gate pulse controls the axis of rotation, an error in the pulse phase directly causes an error, and therefore infidelity, of the applied gate. Phase errors may depend on phase stability of microwave sources and arbitrary wave generators and even on the master clock stability [39]. Principal scheme of the timing signals in our setup is shown in Fig. 3.15. Rubidium clock generates a stable 10 MHz signal which is distributed to the instruments for phase synchronization. Microwave sources are additionally synchronized via higher frequency 1 GHz tone to ensure higher phase coherence and stability. To distribute the timing and trigger signals, the first AWG serves as a master clock of the setup and supplies the triggers to other instruments. It includes sequential AWGs, microwave sources (if necessary) and FPGA readout triggers. As it has been mentioned previously, instruments have internal delays between the arrival of the trigger and the beginning of the signal output. Finite signal speed through the microwave cables (∼ 1ft/ns) adds extra delays due to the difference in configurations of charge and flux lines, room-temperature cabling etc. Together with the requirement to time the pulses and readout on different lines, the complexity of correctly aligning the triggers one-by-one quickly grows out of reach. Our virtual timing instrument stores the delays of all the generating instruments. It accounts for both internal delays, i.e. the time between the arrival of a trigger and the actual start of

46 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.15: Principal scheme of the devices required to run a multi-qubit experiment, see details below in the text. Black arrows indicate clock synchronization signals, i.e. stable 10 MHz tone (1 GHz tone between the microwave sources, if the model allows). Red arrows correspond to timing triggers. Note that FPGAs could be triggered either by the pulser, or by AWGs depending on the choice of the wiring. the output, and external delays due to cabling and passive instruments. Those times can be on the order of hundreds of nanoseconds, though they are ﬁxed by the manufacturer and usually are provided in the corresponding device manuals. Virtual timing instrument is discussed in detail in section 3.4.2.

3.4.1 Pulse alignment organization

We separate experiments into repetitions. A single repetition generally consists of a set of qubit control and manipulation pulses followed by a readout of one or several qubits with the data acquisition on FPGA. Finally, a long time is given for the qubits to decay into their ground states, so the next repetition can start. The length of a single repetition is called repetition time and deﬁnes a rate at which the experiments can be performed. As most of the time in a single repetition is occupied by the qubits thermalizing to their environment, repetition time varies greatly depending on the decay (T1) time of qubits in the working sample. We introduce a time reference within a single repetition called a ﬁxed point in order to easier synchronise the control pulses and readouts. All the gates (control pulses) are then stacked

47 CHAPTER 3. EXPERIMENTAL TOOLBOX

Figure 3.16: Principal timing scheme for a single experiment repetition. before the fixed point with the last pulse ending at the fixed point, and the readouts start at the fixed point (with a small delay specified to avoid overlapping). Figure 3.16 demonstrates the described time structure for each experiment repetition. Unlike the natural idea to use ’zero-time’ point of the repetition as a reference and stack the pulses and triggers after it, this way ensures the readout (and tomography pulses if present) happens in exactly the same time of each repetition and does not ’jump’ between them, thus possibly introducing extra phase shifts. In addition, forceful separation of readout and control pulses in time allows for easier calibration of the instrument delays. We will now consider the physical triggers required to align the pulses’ timings with the example of a single-qubit experiment: that is, one resonator readout source, one AWG channel pair and one FPGA. Clearly, in order to properly trigger the readout resonator and FPGA one has to just send those triggers at the fixed point (with the account of instrument delays, see below). AWG trigger has to come earlier, before it starts outputting its programmed pattern for this specific repetition. The length of an AWG pulse sequence being output in a single repetition is defined by pattern length: the length of a digital AWG pattern being output. As the end of the pattern – and the end of the last pulse – has to come at fixed point, the AWG trigger is to be sent at the point in time shifted by the pattern length before the fixed point. The described set of timing triggers for a single-qubit experiment is shown in Fig. 3.17. Different repetitions within the same experimental run may correspond to exactly the same set of control pulses and readouts in order to acquire statistics about averaged values of observables (such as density matrix elements) or in order to reduce noise. Alternatively, the parameters of manipulation and readout sequence may vary across the repetitions in order to realise multi-segment experiments such as Rabi flopping or measure T1 decay of the qubit. In this case one has to program multiple segments on an AWG with each segment containing a pattern (with the same length of ”pattern length” timing parameter) with a pulse sequence for a specific repetition. Ordinarily, both purposes are used within the same experimental sequence. In this case the AWG loops through the segments in an internal loop, and then an external loop is used to average the results. This approach in contrast to averaging a single segment multiple times before moving to the next one allows mitigating various slow noise contributions.

48 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.17: Principal timing scheme for a single experiment repetition including triggering pulses and markers. Internal trigger delays of instruments and natural delays due to wiring are not taken into account on this schematic picture.

Finally, more complicated sequences involving multiple consequent readouts together with manipulation pulses (such as the one required to measure the effective temperature of qubits, see Chapter6) also may be realized. While based on the same general idea (we prefer to choose the fixed point to be the beginning of the first readout), they require a custom marker function which specifies the additional set of triggers to be sent to the instruments. In this case multiple segments of AWGs and multiple readout triggers would be output during the same repetition.

3.4.2 Instrument delays and virtual timing instrument

Instrument delays – It is not enough to just send the trigger to a device at the time it has to start outputting, as devices have some internal delay times. Those times can be on the order of hundreds of nanoseconds, though they are ﬁxed by the manufacturer and usually are provided in the corresponding device manuals. Together with the requirement to time the pulses and readout on diﬀerent control lines, the complexity of correctly aligning the triggers one-by-one taking into account the instrument delays, delays due to the circuitry and cables quickly grows out of reach. In order to make the process semi-automatic we are using a virtual timing instrument (VTI). The purpose of VTI is to serve as an interlayer between the software pulse generation libraries and the physical triggering. Within the pulse generating software (PulseGen) one can then program the pulses and readout as they should appear at the processor inputs, while the VTI will account for the delays described above and send the triggers and markers asynchronously at the correct times. An important property of the timing instrument is its ability to accept both positive and negative time delays. It allows shifting the readout/acquisition pulses to earlier or later if needed, but also accounting for longer cabling or slower ’trigger-output’ turnaround. If one requires to send the trigger to a device earlier with a cumulative negative physical delay, timing instrument simply sends all other triggers later making use of the fact that the precise real time of the experiment repetition is of no importance. Since the VTI is the layer of transition to an abstract PulseGen software with the ignorance

49 CHAPTER 3. EXPERIMENTAL TOOLBOX of physical connections, it also stores the map of the experiment’s triggering connections and has to be calibrated every time the physical wiring and especially the trigger connectivity changes. As an example, the readout marker to a resonator MW source can be sent via a pulser or via an AWG, and the delay time of that marker would depend on the specific type of the connection. Updating the VTI delay parameters to ensure the pulse alignment under our experimental scheme with a fixed point is relatively straightforward. First, multiple AWGs are calibrated so they end the outputting at the same time. Program- ming a single square pulse on each channel and checking the alignment with an oscilloscope achieves the goal. Resonator readout pulses (readout markers) are aligned in the similar fashion; we target a time delay with the safety interval of 5-20 ns between the end of the AWGs’ pulses and the beginning of the readout pulses. Finally, acquisition triggers sent to FPGAs are aligned so the acquisition starts at the same time as the readout signals arrive to FPGAs. As electronic delay times due to the finite speed of light in the wiring (about 3ns/m in vaccum) are getting comparable to the delay between pulses, the outputs of AWGs and the readout resonator sources are to be aligned not at the devices’ outputs, but at the input to the cryostat. Otherwise a qubit signal going through an extra RT high power input amplifier with extra cabling may be shifted by tens of nanoseconds, which exceeds the safety margin interval between the pulses. While the same issue technically can appear at the cold stage of a cryostat, one does not expect extra meters of cables on the MC plate wiring or on the on-chip control lines, so the timing shifts would not exceed several nanoseconds. However this reinforces the necessity of the ’safety’ delays between different pulses and pulses and readout. A pulser is a very steady instrument referenced by a precise 10 MHz signal, and we use it to control the general timing of our setup, such as marking the start of a new repetition. However it is not suitable for more complex tasks such as outputting two readout markers per repetition with a sweeping delay between them, as required for Chapter6. In this case we use AWG trigger and marker outputs to trigger other AWGs, readout MW sources and FPGAs. As we acquire the readout signal via heterodyne detection [15], it is important to have the phase of the readout signal fixed and constant across the repetitions. We register the down-converted resonator signal at the fixed IF frequency of 25 MHz; the resonator and down- conversion sources are phase-locked. However their relative phase still precesses at this fixed 25 MHz rate according to the fixed frequency difference. A natural way to fix the relative phase across the repetitions is to choose the repetition time commensurate with 25 MHz (or proportional to 40 ns), so the readout always starts from the same phase on this down-converted 25 MHz signal.

3.4.3 PulseGen

Once the timings have been calibrated, aligning multiple pulses for different qubits can be done in software by the means of pulse generating libraries (PulseGen). As all the pulses happen before the fixed point and end at it, the specified pulse sequence is stacked backwards in time from the last one and to the first one from the fixed point and into the past. Therefore, adding

50 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.18: This figure demonstrates adding a new pulse P4 to the set of three pulses P1, P2 and P3 in the pulse generating software. Notice how even though the pulse is added to the end of the sequence, it affects the timings of all other pulses, as the last pulse always ends at the fixed point (see Sec. 3.4.1). a pulse to the end of a sequence will actually shift all the other pulses back by the length of this added pulse plus the specified inter-pulse interval. Our PulseGen library is custom-written and Python-based and allows programming multiple AWGs with the specified sequences of pulses for the specified number of segments. PulseGen has a few standard pulse shapes (square, Gaussian, Derivative Removal by Adiabatic Gate (DRAG) [11]) and supports pulse de-embedding to counter for the line response (see details in Chapter5). In addition, all the basic parameters such as pulse lengths, intermediate frequencies, inter-pulse delays and others can be changed according to the experimental requirements. For example, changing the IF frequency to be up-converted to drive a qubit between the pulses within the same sequence (i.e., within the same segment or repetition) allows driving both a Sg⟩ − Se⟩ and an Se⟩ − Sf⟩ transitions with the same microwave source. As the time alignment is handled by the timing instrument with simple scaling, scaling the programming via PulseGen to many qubits is also straightforward to implement. Using a fixed set of gates with pre-defined lengths to operate a quantum processor reduces one extra step of abstraction and allows realizing more complex algorithms via the sequence of PulseGen → VTI → AWG segments → physical triggering.

3.4.4 Readout in time domain

Readout in time domain is based on the same underlying idea as the CW readout: dispersive shift of the resonator frequency depending on the state of the qubit. A single readout in time domain corresponds to one ﬁxed period of time sending a readout pulse through the resonator, receiving it and registering with an FPGA. As described earlier, an FPGA receives a down-converted to IF (25 MHz) signal, digitizes

51 CHAPTER 3. EXPERIMENTAL TOOLBOX it with a fixed sampling rate and applies a finite impulse response (FIR) filter (see Fig. 3.2); heterodyne detection yields both the signal amplitude and phase for each sample, i.e. a point on the I-Q plane. Therefore, a result of a single readout in time domain is a single time trace on the I-Q plane. Choosing the repetition time to be commensurate with the FPGA input frequency ensures that the global phase of readout traces would not change between repetitions. A straightforward way to process a trace to determine the qubit state is to integrate the whole trace, obtaining a single point on an I-Q plane. In the ”ideal noiseless” case one would obtain one of the two I-Q points: corresponding to Sg⟩ or to Se⟩; with real measurements one would observe two clouds of points whose radii are mostly determined by the electronic noise. It also suggests a direct way to define the measurement’s signal-to-noise ratio as the distance between the resonator responses for Sg⟩ and Se⟩ (which we define rg and re) divided by the standard deviation of the points in one of the cloud along the line connecting rg and re. Alternatively, saving the whole trace as a list of points allows observing the time-resolved signal, which can be beneficial for timings and/or error calibration. Note, even if measurement noise is small, a time trace would be a complex trajectory on an I-Q plane and not a single point due to effects such as ring-up cavity time, parasitic capacitances, analog and digital filtering and many others. Finally, machine learning techniques can be applied to enhance the readout fidelity by training a classification algorithm on set of labelled traces for ground and excited states of the qubit [40]. While this is a developing and promising approach with modern hardware allowing distinguishing the traces in real time and applying feedback based on it if necessary, I will stick to the simpler integration method without the loss of generality.

3.4.5 Averaging and normalization

While performing experiments in the continuous wave regime, the problem of low signal contrast can be solved simply by increasing the measurement time: it would decrease the noise proportionally to a square root of the measurement length (assuming uncorrelated noise), and eventually would yield the required contrast. For time domain readout, this method unfortunately does not work. Indeed, as one increases the length of the readout time trace after preparing the system in a certain state to overcome the electronic noise coming from all sorts of instruments along the way, the system will also decay to its ground state after a short period of time (in principle random, but deﬁned by its T1 time), thus limiting the length of the readout time trace which would increase the signal contrast. The way to overcome the noise in time-domain experiments is to repeatedly prepare the system in the same state and repeat the experiment with measuring the time trace several times. While this is a standard and working approach, it poses a serious limitation on the measurements: averaging masks the results of a single experiment and gives only the expectation values of measurement operators, such as density matrix elements. Let us consider an example Sg⟩+Se⟩ ~ √ of a state after a π 2 pulse on a Bloch sphere (a qubit in a state 2 ). After performing a single measurement, one would (without noise) still see either a result corresponding to Sg⟩ state or to Se⟩ state. So we’ll see one of the two side points with 50/50 probability. However, averaging

52 3.4. TIME-DOMAIN MEASUREMENTS out multiple repetitions of the experiment to overcome the electronic noise, we also average the ’true’ signal and get a point in the middle of the two calibration points corresponding to Sg⟩ and Se⟩ states on the I-Q plane. Figures 3.19a and 3.19b demonstrate the noise reduction via averaging for a qubit readout. Even with the averaging caveat, the ability to measure the density matrix elements of a single quantum system is a remarkable achievement as it allows demonstrating the coherent control over a system directly by performing for example Rabi or Ramsey experiments (which will be introduced later) as well as probing and exploring time-resolved quantum phenomena.

(a) Averaged qubit readout (b) Averaged qubit readout

Figure 3.19: I-Q plane hexbin histogram plots of averaged readout for a qubit prepared in three diﬀerent states. The outcomes are colour-coded. The intensity of the colour represents the number of measurement outcomes falling in each bin. Each measurement point is averaged with 16 (a) and 1024 (b) repetitions. Note that while averaging clearly helps reducing the noise, the single measurement features of a π~2 state also get averaged out.

More speciﬁcally, the way to obtain the magnitude of ρee is as follows. Preparing the Sg⟩ and Se⟩ with high precision and measuring many times yields calibration points rg and re on the I-Q plane. Following that, any state we measure will fall on the line between rg and re, and we can determine ρee by using the equation:

ρee = Re [(r − rg)~(re − rg)] . (3.5)

In other words, the ratio of the distances between the point and rg and re calibration points is also the ratio of ρgg and ρee. It is important to note, we are using the complex voltage (or any of its quadratures) but not the absolute value of the signal. Acquiring the signal, averaging and integrating the traces and taking the absolute value could also work, but the normalization of the element on the I-Q plane would pose an extra complication, as the absolute value of the middle point may be outside of the range of the absolute values of the two end points of a line segment (consider the example of 1+i and 1-i). Taking the absolute value before averaging the signal would often fail completely, as in the case when the signal is weaker than noise, the result would represent just the average noise amplitude.

53 CHAPTER 3. EXPERIMENTAL TOOLBOX

Working with heterodyne voltage – a point on the I-Q plane – oﬀers no extra complications even if the noise is greater than the signal. Averaging decreases the noise proportional to the square root of the amount of the repetitions, thus reducing the sigma of a point cloud on an I-Q plane, and eventually makes it suﬃciently small compared to the separation between the

”true” signal points rg and re.

3.4.6 Single shot readout

One of the key features of quantum mechanics, which lies in a stark contrast with classical mechanics, is that repeating the same experiment with exactly identical starting conditions and manipulations may not lead to the same outcome. It means that while in classical mechanics averaging the signal coming out of the experiment would only average the noise, as averaged ’true’ signal and ’true’ signal are the same objects; in quantum mechanics it is not the same: a ’true’ outcome of a single qubit measurement is a pure 0 or a pure 1, while the averaged

’true’ signal corresponds to a ρee density matrix element. So this way of reducing the noise by repeating the experiment erases the information about single outcomes of measurements, and one only has access to the information and properties which are encoded in the averaged dynamics of a quantum system. Consequently, some of the purely quantum effects are lost if one has access only to the elements of the density matrix of the qubit state. A single shot readout allows one to determine the state of a quantum system in a single run of an experiment, thus giving the experimentalist the full possible information about the quantum system. Certain quantum algorithms directly require single-shot measurements for exploiting the contextuality of quantum states and certifying the randomness coming from a measurement (see Chapter4), performing post-selection based on the (single) measurement outcome or performing the feedback or feed-forward loops, like in quantum teleportation. The key to the single-shot readout of a quantum system is increasing the readout signal- to-noise ratio up to the level so that two readout traces corresponding to the Sg⟩ and Se⟩ states of the qubit are distinguishable in a single run. As we process a time trace by integrating its values first, for the resulting points on an I-Q plane it means that standard deviations σ of point clouds are (significantly) smaller than the distance between the centres of the clouds (see Fig. 3.20). Misinterpretation of an outcome due to not high enough signal contrast is the first obvious error source of the single shot readout. Equipment calibration — Due to strict requirements on SNR with no possible aid from averaging of any sort, equipment calibration becomes the key for achieving high fidelity for single shot readout. First, well-calibrated timings decrease the decay errors due to extra waiting time between the manipulation and readout pulses or loss of signal contrast due to a possible delay between the start of the cavity readout pulse and the start of the acquisition. Note, these errors (including the case when the readout and manipulation pulses are overlapping to some extent) are also being averaged out in the averaged time domain measurement, which makes single shot much more sensitive to them. Noise of the amplification chain plays a key role and provides the dominant contribution to

54 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.20: I-Q plane hexbin histogram plot of single shot readout for a qubit. The outcomes are colour-coded with blue corresponding to the qubit prepared in the ground state, and red – in the excited. The intensity of the colour represents the number of measurement outcomes falling in each bin. Unlike the averaged case, each single point is a result of a single measurement, high contrast of which allows distinguishing the outcomes. the ’noise’ part of the signal-to-noise ratio. In case of the absence of a quantum-limited amplifier such as JPA or TWPA, the noise of the readout should be limited by the noise generated by the transistor-based HEMT amplifier. Making sure that none of the successive RT or IF amplifiers contributes significantly to the total noise budget becomes important for the proper calibration of a single shot readout. Similarly, with the presence of a quantum-limited amplifier (a JPA in our setup), the dominant noise contribution should be coming from it. Note, the HEMT amplification settings may on purpose be lowered in the presence of well-calibrated JPA, as decreasing the HEMT gain might also decrease the added HEMT extra noise. An automatic routine to measure the single shot readout SNR is useful for this and successive steps as it greatly aids in finding the best working points for the amplifiers and readout parameters. Finally, the readout pulse power and duration are two parameters to be calibrated. While they should be calibrated for all measurements, including continuous wave and averaged time- domain measurements, it is not as critical for those as increasing the averaging (or the readout time) would yield the desired SNR. Moreover, higher SNR would lead to higher readout fidelity: with higher SNR one can acquire a readout trace for a shorter period of time (i.e. do a shorter readout pulse), thus suppressing the errors due to the internal T1 and cavity-induced decay. And therefore we want to acquire for as short time as possible to suppress the decay as much

55 CHAPTER 3. EXPERIMENTAL TOOLBOX as possible. For averaged time-domain measurements the cavity power and readout duration are also to be calibrated. The good way to do it is having a metric such as SNR, more specifically a distance between re and rg divided by the averaged noise for a fixed amount of averages. As the noise of a point divided by the rg – re contrast also determines the noise of ρee measurement, it is important to increase the SNR as much as possible. For the averaged time-domain readout increasing the readout duration to be comparable to a T1 decay time of a qubit or even longer still increases the overall signal-to-noise ratio of the readout. Indeed, as long as the averaged trajectories corresponding to Sg⟩ and Se⟩ are still resolvable, integrating the readout trace for longer still increases the resolution. Calibration of the resonator power is done similarly; one might note that the readout power for the pulsed (time-domain) regime is generally much higher than for spectroscopic measurements, as one does not need to abide by the ’less than one photon in the cavity’ rule. However increasing the resonator power beyond a certain threshold leads to breakdown of dispersive approximation, introduces measurement-induced backaction2 or non- linear behaviour of the resonator, in which case the readout SNR (and fidelity) would start decreasing again. For the single shot the situation is drastically different, and generally the readout time is much shorter. There are two main contributions to the measurement noise which come from different sources. The first one is the electronic noise of measurement devices. And identical to the averaged or CW case, the longer we integrate the more this noise gets suppressed. However in the single shot case a decay (or excitation) of the qubit during readout affects the readout in entirely different way, thus making it a prominent noise source: if during the readout the qubit has changed its state, it may lead to mislabelling of its initial state even if the electronic noise is greatly suppressed. And of course, this new noise source affects the readout quality stronger as our readout length increases, so it is beneficial to keep the readout length as short as possible. It is important to note, in the averaged measurement case this effect gets cancelled out by the calibration measurement, as the same fraction of the Se⟩-state population of the qubit decays over the course of the measurement no matter what the population was. Naturally for the situation with two noise sources with conflicting requirements, there is a sweet spot in the readout length. Its numerical value is dependent on many parameters of the actual chip and measurement hardware, but generally it is a small fraction of T1 for a qubit, and therefore is much shorter than the optimal length for averaged readout. In addition, higher signal-to-noise ratio allows further decreasing the readout length and getting rid of extra infidelity associated with qubit decay. Alternatively, decreasing the readout power suppresses the backaction. And lastly, the points between the classification regions and far outside of the clouds come from the events such as decay or excitation during readout, where the trace jumps from the g-trace to e-trace or even get thrown out of the qubit subspace. Applying supervised machine

2In this and following sections under measurement-induced backaction I mean excitation of the qubit by the readout tone, hybridising the resonator and qubit states due to the breakdown of dispersive approximation and similar adverse eﬀects, rather than the expected collapse of the wavefunction.

56 3.4. TIME-DOMAIN MEASUREMENTS

(a) Single shot readout (b) Single shot readout

Figure 3.21: I-Q plane hexbin histogram plot of single shot readout for a qubit. (a) Reference plot with the qubit being prepared either in Sg⟩ or Se⟩ state. (b) The qubit is prepared in the superposition π~2 state. Note the key diﬀerence from 3.19a and 3.19b: the superposition state does appear as two clouds of points, reproducing the probabilistic outcomes of being projected either to Sg⟩ or to Se⟩. learning techniques [40] helps calibrating out these events and judge the states based on traces, thus increasing the ﬁdelity of readout. The downside is extra resource cost compared to a simple method based on the integration and separating regions on the I-Q plane.

3.4.7 Three-level single shot readout

As we have discussed earlier, a transmon is actually not a qubit, but a multi-level system with full controllability of higher levels. Reading out these higher levels in single shot regime brings in an extra complication, which we will consider on the example of three-level single shot readout. Three-level readout with high fidelity is a key element to a number of protocols, enabling things such as violation of Klyachko–Can–Binicioglu–Shumovsky (KCBS) inequality demonstrating contextuality of quantum mechanics [41, 42]. Contextuality combined with a single-shot nature of the readout also serves as a basis for a certified quantum random number generator (QRNG) (see Chapter4). Single shot here is essential, as it is exactly what gives us the access to this quantum randomness, as averaging gets rid of it altogether. Conventionally, we measure the dispersive readout with the resonator probe tone at the frequency corresponding to the ground state of the qubit. However it leads to both Se⟩- and Sf⟩- transmissions being very small, as they are far detuned. It mandates using a cavity readout tone frequency as an extra optimisation parameter and choosing the frequency closer to the Se⟩-state frequency of the cavity, as that will not suppress Se⟩- and Sf⟩- state transmission amplitudes so much and will actually make them more distinguishable. An example of readout plots with different resonator tone frequencies is shown in Fig. 3.22. Figure 3.22a shows a normal readout with a good Sg⟩-Se⟩ contrast, but little to no Se⟩-Sf⟩ contrast. In Fig. 3.22b the readout frequency is detuned from being resonant to Sg⟩-frequency to in- between Sg⟩- and Se⟩- frequencies, which enhances the contrast between Se⟩ and Sf⟩ allowing to distinguish them in a single shot. This also serves as a demonstration of an advantage of

57 CHAPTER 3. EXPERIMENTAL TOOLBOX heterodyne detection, as we can not only see the diﬀerence of the signal amplitude, but also detect its phase.

(a) Single shot readout (b) Single shot readout

Figure 3.22: I-Q plane hexbin histogram plots of single shot three-level readout for a qutrit. The outcomes are colour-coded. The intensity of the colour represents the number of measurement outcomes falling in each bin. (a) The readout frequency equals to the resonator’s peak corresponding to the Sg⟩ state of a qutrit, i.e. a standard readout. Note the contrast between the Se⟩ and Sf⟩ states is virtually absent. (b) The readout frequency is oﬀset by 7.4 MHz and is closer to the Se⟩-state resonator frequency, thus leading to much better contrast between Se⟩ and Sf⟩. Both readouts are not optimised with respect to the readout length and pulse power, thus one can see a lot of decay during readout (especially clear on 3.22b)

Full readout optimisation routine includes three parameters: resonator readout power, resonator readout frequency and readout duration and uses the fidelity of three-level single shot readout as a target value to optimise. For the relatively short-lived qubit we have employed to generate certified random numbers higher powers and shorter readout lengths turned out to be preferred, as it greatly suppresses the decay during readout. In addition, readout tone off-resonant with any of the dispersively shifted resonator frequencies (corresponding to Sg⟩, Se⟩ or Sf⟩ qubit states) allowed driving the resonator with a stronger tone (and thus receiving more signal) without hybridising the state during the readout to see the measurement backaction. A technical complication arising from a narrow-band character of the JPA we are using – the amplifier has just a few MHz bandwidth with 30 dB amplification, which seems to be optimal for our system’s parameters – can easily be addressed by pre-calibrating the JPA parameters for a grid-like set of working points, which allows to tune the readout frequency and the JPA maximum amplification point together during the calibration process.

3.4.8 Rabi measurement

Demonstrating the ability to coherently control the qubit, the ﬁrst and most basic time-domain experiment is the observation of Rabi ﬂopping. The protocol for it is simple: we apply a coherent microwave pulse at the qubit transition frequency to the transverse (charge) qubit drive line followed by a measurement pulse applied to the coupled resonator (see Fig. 3.24).

58 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.24: The experimental protocol for Rabi and Ramsey experiments. Each protocol is repeated multiple times with various magnitudes of the applied pulse α (Rabi) or with various delays τ between consecutive π~2 pulses.

The microwave pulse rotates a qubit on the Bloch sphere; this rotation results in sinusoidal oscillations between the ground and excited states of the qubit. In order to observe that, let us recall the Hamiltonian of a driven qubit discussed at the end of Chapter2.

h̵ H = ω σ + hA̵ cos (ω t + φ ) σ + hA̵ (t)σ (3.6) 2 q z x x x x z z

Ignoring the σz coupling term (not sending any signal through it) and sending a pulse at the frequency resonant to the qubit’s energy splitting leads to a single time-independent rotation in the frame rotating with the frequency of the drive, whose phase is dependent on the phase of the drive, see Eq. 3.7.

1 Figure 3.23: Rabi oscillations on a Bloch sphere. H = ωqσz + Ax cos (ωxt + ϕx) σx 2 The rotation vector Ω depends on the phase of the i S U = exp ω tσ ω = ω (3.7) drive, thus allowing us to apply diﬀerent gates. 1 2 x z x q 1 H′ = A cos ϕ σ′ + sin ϕ σ′ 2 x x x x y

While the Hamiltonian of a driven qubit in the rotating frame suggests that one can apply longer pulses to rotate the qubit further on the Bloch sphere, we prefer to ﬁx the pulse length and sweep the pulse amplitude instead. It means that recalibration of a π and π~2 pulses would not change their lengths, which makes aligning multiple pulses on multiple qubits easier in the future. An example of Rabi ﬂopping if shown in Fig. 3.25. Heterodyne detection yields the information about both I and Q quadratures of the readout signal, which we represent by a complex number. For spectroscopy measurements we were interested only in its amplitude; in the Rabi measurement case it wouldn’t work. The reason for that is easy to see: if a point undergoes in time sinusoidal oscillations between two points

59 CHAPTER 3. EXPERIMENTAL TOOLBOX on a complex plane, its modulus also undergoes sinusoidal oscillations if and only if the two points lie on the same line with (0, 0). Since we would like to fit the Rabi oscillations to a sine and obtain a period to calibrate the pi-pulse, we should choose one of the two quadratures (or any rotation) as input to the fitter. Since this choice is arbitrary at this point, the chosen quadrature could contain little to no contrast; in this case one should fit another (orthogonal) quadrature and observe better contrast. Since Rabi flopping experiment is the first time-domain calibration measurement in the calibration sequence, Fig. 3.25 has a voltage Rabi experiment quadrature as the y-axis. Indeed, before the data 0.015 π pulse is calibrated it is not possible to pre- − fit pare the excited state of a qubit and learn 0.020 − its measurement response re (see Sec. 3.4.5) 0.025 − to normalize the response with respect to the 0.030 − Sg⟩ and Se⟩ state (or to measure the ρee den- 0.035 sity matrix element). However for all consec- V quad − 0.040 utive calibration measurements and protocol −

0.045 applications it becomes possible to perform − the normalization and plot ρee or < σz >. 0.050 −

0.0 0.1 0.2 0.3 0.4 Source amplitude 3.4.9 Ramsey experiment Figure 3.25: Rabi flopping experiment. Fitting to While spectroscopy works well to roughly es- an oscillation function yields an amplitude for a π timate the frequency of a qubit and is suffi- and π~2 pulses. cient to obtain Rabi oscillations and roughly calibrate the π and π~2 pulses, Ramsey sequence allows obtaining qubit frequency pre- cisely3. Shown in Fig. 3.24, the protocol consists of applying two inverse pi-half pulses with a delay τ between them, and sweeping τ as a parameter. When we do the spectroscopy, the qubit and the readout resonator coupled to it are driven simultaneously. And therefore the drive applied to the resonator via the resonator-qubit coupling may shift the resonant frequency of the qubit (see discussion in Sec. 3.3.3), with the single-photon regime not necessarily available. Let us consider the same driven qubit Hamiltonian as in the previous subsection (and also ignoring the longitudinal term), but now remembering that we can turn the charge (σx) drive on or off.

3Both names – Rabi flopping (oscillations) and Ramsey experiment – highlight the inheritance of these basic calibration methods from the field of nuclear magnetic resonance (NMR). Just as Rabi flopping is named after Isidor Isaac Rabi, one of the fathers of NMR, Ramsey experiment was originally proposed by Norman Ramsey for the same field in 1950 [43]. Both physicists were later awarded Nobel prizes for their discoveries.

60 3.4. TIME-DOMAIN MEASUREMENTS

̵ h ̵ H = ωqσz + I(t) · hAx cos (ωxt + φx) σx, (3.8) 2 Where I(t) is an indicator function being zero if the drive is oﬀ, and equal to unity if the drive is on (a pulse is applied). Similarly to the Rabi case, consider a general transformation of the Hamiltonian (3.8) into the rotating frame with

i U = exp (ω t + φ )σ , 1 2 x x z but now allowing the drive frequency ωx to be diﬀerent from the qubit splitting ωq. The Hamiltonian in the frame rotating with the drive frequency reads ̵ ̵ ′ h ′ h ′ H = I(t) · Axσ + δωσ , (3.9) 2 x 2 z

where δω = ωq − ωx is the detuning of the excitation driving and we have used the rotating wave approximation under the assumption Ax ≪ ωq. Note that the drive term still has the I(t) multiplier indicating if the drive is on or oﬀ. Diﬀerently to the Rabi case however, we have a second term (always present), which makes the Bloch sphere of the qubit in this rotating frame precess along the central axis.

Ramsey experiment Ramsey experiment T2

1.00 data 1.00 data ﬁt ﬁt 0.75 0.75

0.50 0.50

0.25 0.25

z 0.00 z σ σ 0.00

0.25 0.25 − −

0.50 0.50 − −

0.75 0.75 − −

1.00 1.00 − − 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Delay between pulses, µs Delay between pulses, µs

(a) Pulse frequency calibration (b) Decoherence eﬀects

Figure 3.26: Two applications of Ramsey sequence. (a) Allows to calibrate the qubit frequency with high precision. We add extra detuning to make ﬁtting more robust and decrease the length of the delay τ between the π~2-pulses. (b) Longer Ramsey sequence allows observing the eﬀects of qubit’s decoherence, see more details in Sec. 3.4.11.

To picture the way the protocol together with the expressions above work, let us consider the state of the qubit in the frame rotating with the frequency of the drive. After the ﬁrst π~2 pulse is applied, the qubit state is rotated onto the equator of the Bloch sphere, and the drive

61 CHAPTER 3. EXPERIMENTAL TOOLBOX

term in Eq. 3.9 is turned off. At the same time, due to the second σz present term, the sphere revolves along the central axis with the angular speed proportional to the detuning between the drive and qubit transition frequencies δω. Therefore, position of the qubit on the equator of the Bloch sphere before the second π~2 pulse is applied depends linearly on both the detuning and time delay τ. It leads to sinusoidal oscillations of the qubit state between Sg⟩ and Se⟩, fitting which one could obtain the detuning between the drive and the qubit frequency. An example of Ramsey experiment is shown in Fig. 3.26a. Since the spectroscopy might happen to be close to the real frequency of the qubit, one would not be able to tell apart ”positive” from ”negative” detuning. To counter this problem without the necessity to perform a long Ramsey experiment, where the decoherence starts playing a role, we program the pulses with a known extra detuning and subtract it from the fitted values.

3.4.10 Rabi chevron

While for quantum computing and pulse calibration purposes we prefer to change Rabi pulse amplitude while having the length fixed, it is of course also possible to observe Rabi flopping while varying pulse length. As the only thing determining the overall rotation angle on the Bloch sphere is the time integral of the pulse amplitude (see Eq. (3.9)), the picture would be entirely identical to Fig. 3.25. However, varying the length with fixed pulse amplitude allows us to observe another consequence of the driven qubit Hamiltonian and see the effects of detuned (non-resonant) qubit driving. Let us recall the calculations from the previous sub-chapter and Eq. (3.9), but now consider the detuning δω to be comparable to the charge driving strength Ax.

h̵ h̵ H′ = A σ′ + δωσ′ (3.10) 2 x x 2 z Evidently, adding a detuning introduces an extra rotation of the qubit (or the Bloch sphere) along the z-axis. The rate of this extra rotation is proportional to the detuning; the full rotation vector is now (Ax, 0, δω). It leads to a tilted rotation axis on the Bloch sphere, which results in two effects: first, the amplitude of the oscillations decreases as the detuning increases (the lowest point hit by the circle gets higher as the rotation axis gets closer to σz). And second, the rate of the oscillations increases with no change in the drive amplitude A . To show that, » x 2 2 one can define the full Rabi frequency ΩR = δω + Ax. This effect – increase in the full Rabi frequency by the means of detuning the drive tone while keeping its amplitude relatively low – will prove to be useful in Chapter5. Demonstrating the described effect is the so-called ’chevron’ pattern shown in Fig. 3.27. Named due to the visible resemblance to the military marks of distinction, it shows coherent oscillations between the Sg⟩ and Se⟩ states of the qubit as a function of the drive pulse frequency (or detuning) and its length. Note that varying the pulse amplitude along the y-axis of the figure would lead to the change in the rotation axis and a non-linear change in the overall

62 3.4. TIME-DOMAIN MEASUREMENTS

Figure 3.27: A theoretical modelling of the Rabi chevron pattern. The colorbar corresponds to the probability to ﬁnd the qubit in an excited state. Detuning of the drive frequency (x-axis) from the qubit’s energy splitting leads to faster Rabi oscillations yet being tilted on the Bloch sphere, thus not reaching the Se⟩ state fully.

rotation amplitude, resulting in entirely diﬀerent pattern.

3.4.11 T1 and T2 times

One of the key properties of a qubit is its coherence, which is conventionally described by two times – T1, or qubit’s lifetime (relaxation time), and coherence time T2, also called dephasing time. Inherited from the ﬁeld of NMR, the relaxation time T1 is a characteristic timescale on which the qubit loses its state (or excitation) and decays into the ground state; the dephasing time T2 is characteristic to the loss of coherence (i.e. relative phase) between the ground and excited states [44].

More formally, T1 is the exponential rate of decay of the ρee element of a qubit’s density matrix, and T2 is the exponential rate of decay of ρge and ρeg oﬀ-diagonal elements. Evidently, the diagonal element can not decay with oﬀ-diagonal elements intact, so T2 has some limiting contribution from T1 (T2 ≤ 2T1)[6, 45, 46]. There is no inverse limitation, as a qubit may lose its coherence quickly due to pure dephasing while having its relaxation time T1 orders of magnitude longer.

Following the deﬁnition of T1, one can prepare any qubit state which is not Sg⟩ to measure it. Applying a π pulse as shown in the protocol for T1 measurement in Fig. 3.28 increases

63 CHAPTER 3. EXPERIMENTAL TOOLBOX

Figure 3.28: The experimental protocols for T1 and T2 measurements. (T1) For each segment a π pulse is applied, followed by a waiting time τ before taking a measurement. To maximize the measurement precision, τ is to be swept from zero times to several (3-4) T1s. (T2) While the protocol is identical to the Ramsey sequence outlined in Sec. 3.4.9, the detuning of the pi-half pulses in T2 measurement is smaller and the waiting times τ are swept in much larger range to observe the decay of the Ramsey fringes. the measurement contrast, though is not necessary. Moreover, even if the excitation pulse is detuned from the qubit’s frequency, the protocol will still work as the information about T1 is contained in the rate of the decay curve with time rather than its amplitude. A typical decay curve for a decay time measurement is shown in Fig. 3.29a.

T1 experiment Ramsey experiment T2 1.00 data 1.00 data ﬁt ﬁt 0.75 0.75

0.50 0.50

0.25 0.25 z z

σ σ 0.00 0.00

0.25 − 0.25 − 0.50 − 0.50 − 0.75 − 0.75 − 1.00 − 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Delay before readout, µs Delay between pulses, µs

(a) (b)

Figure 3.29: Typical plots for a T1 and T2 measurements. On the y-axis for both plots is the σz expectation value.

Another application of the Ramsey experimental sequence allows determining the qubit’s decoherence time T2. As the Ramsey oscillations arise from the qubit’s state rotating on the equator of a Bloch sphere (see Sec. 3.4.9), decoherence of the qubit leads to the exponential decrease in their visibility as loss of the phase coherence can be understood as ”dilution” of the qubit’s state along the latitudes of the sphere. Therefore, the experimental protocol for T2 measurement coincides with the Ramsey protocol (see Fig. 3.28), yet with lower detuning and longer time delays τ to allow the decoherence to have an eﬀect. Figure 3.29b demonstrates a

T2 measurement with the corresponding ﬁt with the ﬁt function of

64 3.4. TIME-DOMAIN MEASUREMENTS

e−t~T2 cos(δωt) (3.11)

While both T1 and T2 experiments are straightforward to implement, one has to take some care with timings for both protocols. Normally for a repetition the AWG pattern length is relatively short, while the most of the time (several T1s) is dedicated to the qubit’s relaxation into the ground state. However to accurately determine the T1 exponential decay of the measurement curve (Fig. 3.29a), one has to make the pattern length already to be several T1s long. And therefore the repetition time has to be increased twofold to avoid the qubit being still excited from the previous repetition at the beginning of a new one. Overall, it is the knowledge of T1 and T2 which allows choosing the appropriate repetition times for all other experimental or computational protocols.

Both T1 and T2 times for a qubit greatly depend on the qubit’s bias frequency. Main systematic reasons include Purcell eﬀect arising from the qubit’s coupling to its readout resonator

(mostly limiting T1)[47, 48] and increased sensitivity to ﬂux noise if tuned far from the qubit’s symmetry point (which mostly limits T2). In addition, non-systematic reasons such as couplings to spurious two-level ﬂuctuators and even avoided crossings with hybridisation at certain frequencies may greatly reduce coherence times. Finally, both properties have been reported to not be stable in time, exhibiting changes of up to 50 % and more [49]. While the reasons are not currently well understood, it suggests the parameters such as Γ↓ and Γ↑ are not constant in time, and one or more thermal baths a qubit is coupled to (see the results of Chapter6) are not in a static equilibrium.

3.4.12 Qubit active reset

The repetition rate can be greatly enhanced (i.e., the repetition time is reduced) by imple- menting an active qubit reset protocol as an alternative to waiting multiple T1 times for the qubit to thermalize. It becomes particularly important as the coherence of qubits increases, as the length of the protocol pulses and readout becomes smaller and smaller fraction of the total repetition time. In addition, employing an active reset protocol allows decreasing the spurious population of the excited state, thus increasing the quality of the qubit’s initialization and mitigating the fact that the qubit’s eﬀective temperature for an actual experimental setup is always found to be well above the temperature of the mixing chamber plate of a dilution cryostat (see Chapter6 for more details). While our hardware based on FPGA to detect the state of a qubit allows realizing real-time feedback and reset the state of the qubit based on the result of the measurement, we prefer to use a recently proposed unconditional reset which does not require real-time feedback and is performed with an extra microwave drive instead [3, 50]. Apart from being easier from the programming and technology point of view, the protocol ”outperforms existing measurement-based and all-microwave driven reset schemes in speed and ﬁdelity” [50]. We follow the guidelines of [3] to implement the reset protocol.

65 CHAPTER 3. EXPERIMENTAL TOOLBOX

The idea is based on using the higher levels of transmon to enable the transition which is forbidden from the computational subspace by selection rules. After putting the excitation of the qubit (population of Se⟩ state) out of the computational subspace onto the Sf⟩ state of the transmon (πef pulse), performing the π pulse on the Sf⟩S0⟩ - Sg⟩S1⟩ transition of a joint resonator-transmon system allows transferring the excitation into resonator, where it quickly decays as the resonator lifetime is much shorter than the lifetime of the qubit. The calibration of the reset π pulse consists of the steps generally similar to the standard pulse calibration of qubit transitions discussed in the sections above, but with a few diﬀerences. First, pulsed spectroscopy is used to determine the frequency of the Sf⟩S0⟩ - Sg⟩S1⟩ transition

(see Fig. 3.30a). The protocol for the spectroscopy consists of applying two consecutive πge and πef pulses to transfer the qubit’s population to the Sf⟩ state followed by a long square spectroscopic pulse to observe the transition.

Pulsed spectroscopy – reset f0 g1 oscillations | i − | i 0.8 data 0.2 data ﬁt 0.7 0.1

0.6 0.0

0.1 0.5 − e P 0.2 0.4 −

0.3 0.3 −

Voltage quadrature, a.u. 0.4 0.2 − 0.5 − 0.1

0.18 0.19 0.20 0.21 0.22 0.0 0.1 0.2 0.3 0.4 Reset drive IF, GHz Pulse length, µs

(a) (b)

Figure 3.30: Calibration sequence for a reset pulse. (a) Pulsed spectroscopy of the reset transition. On the x-axis is the intermediate frequency supplied to the IQ mixer by an AWG. (b) Rabi-like damped oscillations allow determining the length of the reset π pulse.

To determine the amplitude of the reset π pulse, we perform a Rabi-like experiment similar to Sec. 3.4.8. However, as the qubit starts the repetition in the ground state, we also add

πge and πef pulses to the beginning of the protocol. As changing the pulse amplitude for the reset transition leads to the AC-Stark shift of the transition [3], the amplitude of the the pulse used in the Rabi experiment should be the same as used in the spectroscopic measurement. Moreover, it mandates using a square pulse and performing Rabi oscillations in time rather than in amplitude (see Fig. 3.30b). Rabi oscillations are heavily damped as the relaxation time of Sg⟩S1⟩ state to Sg⟩S0⟩ is comparable to the length of the π pulse on the transition. Note that is changing the amplitude of the reset pulse (to decrease its length, for example) requires re-calibrating its frequency by performing the pulsed spectroscopy again. Reset sequence may also be used to reset the state of a qutrit, only in this case one would require applying two sequences: π pulse on the Sf⟩S0⟩ - Sg⟩S1⟩ followed by the waiting time to

66 3.4. TIME-DOMAIN MEASUREMENTS allow Sg⟩S1⟩ to decay to Sg⟩S0⟩ gets rid of the population of the Sf⟩ state, and the standard sequence (πef - π pulse on the Sf⟩S0⟩ - Sg⟩S1⟩ - waiting time) eliminates the population of the Se⟩ state. Figure 3.31 demonstrates the quality of the ground state preparation without and with the reset sequence for a qutrit for a repetition time being shorter than the T1 of the qubit.

(a) Single shot readout (b) Single shot readout

Figure 3.31: Single shot readout of a qubit prepared in the ground state. The repetition time of 6.4 µs is less than the relaxation time of the qubit. (a) No reset sequence is applied. A considerable amount of time the qubit is measured to be in the Se⟩ state. (b) An application of the reset sequence greatly enhances the quality of initialization.

67 BIBLIOGRAPHY

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72 CHAPTER 4

QUANTUM RANDOM GENERATOR CERTIFIED WITH THE KOCHEN-SPECKER THEOREM

Random numbers are required for a variety of applications from secure communications to Monte-Carlo simulation. Yet randomness is an asymptotic property and no output string generated by a physical device can be strictly proven to be random. In this Chapter we report an experimental realization of a quantum random number generator (QRNG) with randomness certified by quantum contextuality and the Kochen-Specker theorem. The certification is not performed in a device-independent way but through a rigorous theoretical proof of each outcome being value-indefinite even in the presence of experimental imperfections. The analysis of the generated data confirms the incomputable nature of our QRNG. This chapter is published in Phys. Rev. Lett. 119, 240501 (2017) [1].

4.1 Certiﬁed QRNGs

While we can consider a mathematical abstraction of a true random number generator and examine its properties, in the physical world we are confined to performing finite statistical tests on the output strings. By applying sets of such tests (like NIST [2] or diehard [3]) we can verify with arbitrarily high probability that the generator is NOT random (if it has failed at least one test), but cannot prove its randomness in the opposite case. As an example, one may construct a pseudo-random number generator which passes all above-mentioned tests while the produced sequence is deterministic and even computable [4]. The impossibility of a rigorous proof of randomness for a finite string generated by a physical device motivates the consideration of more fundamental arguments to support a RNG’s randomness. From this point of view, no classical RNG may be truly random as it is deterministic by the laws of classical mechanics, and may in principle be predicted. A natural foundation to build a RNG would be quantum theory, as it is intrinsically random. However, although quantum mechanics obeys probabilistic rules, the possibility of separating intrinsic randomness from apparent randomness arising from a lack of control or from experimental noise is still under debate [5]. Moreover, while quantum mechanics for a two-level system is described by the same intrinsically-probabilistic measurement rules, one may not strictly prove value-indefiniteness, and hence indeterminism, of its results [6]. A proposal to use quantum systems with higher dimensions to produce quantum random numbers, certified

73 CHAPTER 4. QUANTUM RANDOM GENERATOR CERTIFIED WITH THE KOCHEN-SPECKER THEOREM

“1”

“0”

Figure 4.1: The theoretical protocol of the QRNG certified by the strong Kochen-Specker theorem proposed in Ref. [13]. The protocol is formulated for a spin-1 particle and consists of two sequential measurements. The first measurement is used to initialize the particle in the Sz = 0 eigenstate of the spin operator Sz. The second measurement is performed in the eigenbasis of the Sx operator with the two outcomes Sx = ±1 realized randomly as proven by the Kochen-Specher theorem. The outcome Sx = 0 is never realized in the ideal case but can be used to monitor the quality of the protocol implementation. by value-indefiniteness has been proposed [7] as early as in 2009. In addition, observation that randomness requires incompatible measurements have been made even earlier [8,9] These considerations led to the next advance in quantum number generation: the protocols certified by violation of certain Bell-type inequalities [10–12]. More specifically, through violation of the CHSH inequality one may certify that the observed outputs are not entirely predetermined and write a lower bound on the generating process entropy. Unfortunately, this approach does not allow one to close the gap between this lower bound and true randomness. In addition, the Bell-type certification schemes can be regarded as random expanders rather than generators due to the requirement of “a small private random seed” to operate [10, 13, 14]. Finally, the random number generators certified by Bell inequalities utilize no-signaling assumption and is, therefore, inherently a non-local device which is challenging to use for practical applications. To address this problem, a different approach to QRNG certification based on the Kochen- Specker theorem and contextual measurements has recently been proposed [13]. It does not allow certification of the data in a device-independent fashion like the CHSH inequality, but yields a rigorous theoretical proof of measurements outcomes being value-indefinite even in the presence of experimental imperfections. In this Letter we experimentally realize a random number generator certified by the Kochen-Specker theorem. We use the circuit quantum electrodynamics (QED) as a platform for the physical realization of a QRNG. A superconducting qutrit has been used recently to demonstrate quantum contextuality, the resource underlying the operation of the QRNG [15]. Utilizing high controllability and fast repetition rates for circuit QED devices we reach a bit rate two orders of magnitude higher than previously reported certified random number generators [10, 12, 14].

4.2 Protocol and realization

To realize the protocol shown in Fig. 4.1 we use a superconducting quantum system, called a transmon, coupled to a microwave cavity. The transmon has a weakly anharmonic multi-

74 4.2. PROTOCOL AND REALIZATION level structure [16], and its three lowest energy eigenstates S0⟩, S1⟩ and S2⟩ are used as the logical states of a qutrit (see Fig. 4.2). In the dispersive regime, where the cavity resonance frequency is sufficiently detuned from the qutrit transition frequencies, the qutrit-cavity interaction causes cavity frequency shifts dependent on the populations of the energy eigenstates of the transmon [16]. These shifts, called dispersive shifts, are extensively used for realizing dispersive readout of superconducting qubits and qutrits by measuring microwave transmission through the cavity (for a specific example of the measurement of a qutrit, see Ref. [17, 18]). Manipulations of the qutrit quantum state can be realized with microwave pulses resonant to the S0⟩ − S1⟩ or S1⟩ − S2⟩ transition frequencies and applied to the qutrit through a separate i,i+1 on-chip charge line. In the following we define Rnˆ (φ) as rotations of a quantum state of angle φ about the axisn ˆ in the qutrit subspace spanned by {Si⟩, Si + 1⟩}. In particular, one can realize the following rotations

⎛1 0 0 ⎞ ⎜ ⎟ 12( ) ≡ ⎜ ⎟ Ry θ ⎜0 cos θ~2 sin θ~2⎟ ; ⎝0 − sin θ~2 cos θ~2⎠ (4.1) ⎛ cos θ~2 sin θ~2 0⎞ ⎜ ⎟ 01( ) ≡ ⎜ ⎟ Ry θ ⎜− sin θ~2 cos θ~2 0⎟ . ⎝ 0 0 1⎠

To reformulate the protocol shown in Fig. 1 in terms of energy eigenstates of the transmon we map the eigenstates of the Sz operator to the states of the qutrit as follows

{Sz, −1⟩, Sz, 0⟩, Sz, +1⟩} → {S2⟩, S0⟩, S1⟩}. (4.2)

In the eigenbasis of the qutrit the spin-1 operator will take the form

⎛0 0 0 ⎞ ⎛0 1 1⎞ ⎜ ⎟ 1 ⎜ ⎟ Sz ≡ ⎜0 1 0 ⎟ ,Sx ≡ √ ⎜1 0 0⎟ , (4.3) ⎜ ⎟ 2 ⎜ ⎟ ⎝0 0 −1⎠ ⎝1 0 0⎠ with eigenstates of the operator Sx √ √ ⎛− 2⎞ ⎛ 0 ⎞ ⎛ 2⎞ 1 ⎜ ⎟ 1 ⎜ ⎟ 1 ⎜ ⎟ Sx, −1⟩ = ⎜ 1 ⎟ , Sx, 0⟩ = √ ⎜−1⎟ , Sx, 1⟩ = ⎜ 1 ⎟ . (4.4) 2 ⎜ ⎟ 2 ⎜ ⎟ 2 ⎜ ⎟ ⎝ 1 ⎠ ⎝ 1 ⎠ ⎝ 1 ⎠

For our qutrit encoding the system is initialized in the ground state SSz = 0⟩ = S0⟩ by cooling down the transmon to the base temperature of a dilution cryostat (∼ 20 mK), thus performing the ﬁrst measurement in the protocol shown in Fig. 4.1. The spurious thermal population of the excited states has been measured to be < 1%.

75 CHAPTER 4. QUANTUM RANDOM GENERATOR CERTIFIED WITH THE KOCHEN-SPECKER THEOREM

I LO (a) AWG (b) υ ,υ ,υ ~ 7 GHz Q RF 01 12 c RF υ01 − υ12 ∼ 300 MHz

|2〉

|2〉 υ 12 g |1〉 C JPA C HEMT RT I |1〉

ADC υ01 υc Q LO |0〉 |0〉 20 mK 4 K 300 K qutrit cavity

Figure 4.2: (a) Simplified diagram of the measurement setup. A transmon type multi-level quantum system is incorporated into a 3D microwave copper cavity attached to the cold stage of a dilution cryostat. A magnetically tunable Josephson junction (SQUID) is used to control the transition frequency of the qutrit by a superconducting coil attached to the cavity. Amplitude-controlled and phase-controlled microwave pulses are applied to the input port of the cavity by a quadrature IF (IQ) mixer driven by a local oscillator (LO) and sideband modulated by an arbitrary waveform generator (AWG). The measurement signals transmitted through the cavity are amplified by quantum Josephson parmateric amplifier (JPA), by a high-electron-mobility transistor (HEMT) amplifier at 4 K and a chain of room temperature (RT) amplifiers. The sample at 20 mK is isolated from the higher temperature stages by three circulators (C) in series. The amplified transmission signal is down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated LO, and is digitized by an analog-to-digital converter (ADC) for data analysis. (b) The energy level diagram of a qutrit coupled to a microwave cavity. The transition frequencies of the qutrit and cavity are in GHz while the anharmonicity of the qutrit is ∼ 300 MHz. When the coupling g between the transmon and the cavity is much smaller than their mutual detuning, the system is in the dispersive regime used for measurement of the qutrit.

Our dispersive readout realizes a projective measurement of the qutrit described by three operators: {S0⟩⟨0S, S1⟩⟨1S, S2⟩⟨2S}. In order to perform measurement in the eigenbasis of Sx we followed the standard procedure and performed rotations of the state before and after the dispersive measurement. More speciﬁcally, to measure some arbitrary state Sψ⟩ in the eigenbasis † 01 12 of the Sx operator, we ﬁrst apply M = Ry (π~2) · Ry (π~2) to rotate the state of the qutrit Sψ⟩ before the dispersive measurement

† 01 12 M Sψ⟩ = Ry (π~2)Ry (π~2)(α−1Sx, −1⟩ + α0Sx, 0⟩ + α1Sx, 1⟩) (4.5)

= α−1S1⟩ + α0S2⟩ + α1S0⟩.

During the dispersive measurement the state is projected to one of the energy eigenstates Si⟩ with 2 probabilities described by SαiS . Then we can apply an additional rotation M to make the full procedure equivalent to the measurement described by {Sx, −1⟩⟨x, −1S, Sx, 0⟩⟨x, 0S, Sx, 1⟩⟨x, 1S}. Note that the last rotation does not change the outcome of the measurement and was not implemented in the actual protocol. As the system is initialized in S0⟩ state the measurement will produce outcomes Sx ± 1 encoded as “1” and “0” with equal probabilities while Sx = 0 outcome will ideally never be realized. If outcomes Sx = 0 are detected these traces can be

76 4.2. PROTOCOL AND REALIZATION discarded and will not aﬀect the randomness of the generated numbers in accordance with the recipe of Ref. [13].

4.2.1 Three-level single-shot readout

To distinguish between three different states with high fidelity we use a Josephson parametric amplifier similar to the one described in Ref. [19]. In addition, we set the readout pulse frequency close to the cavity frequency corresponding to the S1⟩ state of the qutrit, which allowed the three possible qutrit states to be well separated on I-Q plane (see Fig. 4.3(a)). The readout frequency was fine-tuned to maximize the three-level readout fidelity. Using the outlined procedure for initialization and measurement we generated 10 Gbit of raw data at a rate of 50 kbit/s (see Fig. 4.3(b) for logical encoding of the resulting states and the correspondence to the spin-1 protocol).

4.2.2 Robustness to noise and imperfections

If the qutrit is prepared in the state Sφ⟩ and we perform a quantum measurement described by the projectors Sψ⟩⟨ψS then Ref. [13] (improved in Ref. [20]) provides the condition to certify the value-indeﬁniteness of the outcomes of the measurements: ¾ 5 3 ≤ S⟨ψSφ⟩S ≤ √ . (4.6) 14 14

In our protocol we take {Sz = 0} state as Sφ⟩ and {Sx = ±1} as Sψ±⟩ (see Fig. 4.1). If our system were ideally prepared in the ground state and all the experimental imperfections were generated only by errors in the microwave control we could estimate S⟨ψ±Sφ⟩S directly as the square root of the probability to obtain the outcomes “0” and “1”. The resulting probabilities to obtain ”0” and “1” were measured as 0.536 ± 0.004 and 0.464 ± 0.004 confirming that the control errors of our setup guarantee value-indefiniteness with high confidence. In reality the actual states of the system before and after the measurement are not described by pure states. The main contribution to the deviation of the probabilities from the ideal value of 1~2 is due to relaxation of the qutrit during the dispersive measurement. As it leads to the misinterpretation of the excited state as being the ground state, we measured greater probability to obtain “0” rather than “1”. Another sources of imperfections are thermal excitation of the qutrit (< 1%), fidelity of gates (> 99%) and misinterpretation of the outcome due to amplifier noise (0.006%). The result of these imperfections may lead to a situation when for some runs the certification condition will not be fulfilled. To provide a confidence low bound for randomness to be certified we conservatively assume that the deviation of the probabilities from the ideal value 1~2 is only due to the runs where the certification condition (4.6) is not valid. Thus, we estimate that only 95% of our generated bits are certified random. As a last step, we address the bias in probabilities of getting “0” and “1” by a standard procedure. For each bit of final data we perform the measurement two times in a row. We

77 CHAPTER 4. QUANTUM RANDOM GENERATOR CERTIFIED WITH THE KOCHEN-SPECKER THEOREM

(a) (b)

Figure 4.3: (a) Hexbin histogram plot of single-shot three-level readout of diﬀerent qutrit states. Red: ground qutrit state; Green – excited state (S1⟩). Blue: second excited state (S2⟩). The intensity of the color represents the number of measurement outcomes falling in each bin. (b) Hexbin plot of the output of the protocol. Shown are logical encoding of the resulting states and the correspondence to the spin-1 protocol. Note, the Sx = 0 state is almost (< 0.1%) never realized. The black lines sketch the boundaries of the classiﬁcation regions.

encode logical “0” and “1” in the physical events “01” and “10” respectively, which have the same probability to occur, and ignore the two other outcomes. It is straightforward to prove that the properties of QRNG will be preserved: new bits will be certified by value indefiniteness and independent from each other. This normalization process yields an unbiased sequence with probabilities of “0” and “1” to be 50% each, which is supported by the obtained 50.001% mean frequency of obtaining the 0 outcome and the standard deviation of 0.1%, which is consistent with the bucket size of 999302 raw bits produced. It also increases certification bound: 99.7% of the final bits are certified random: it is sufficient to have one random physical event in the logical sequence to certify the whole sequence to be random. For more a optimized scheme of data post-processing in order to increase randomness of the raw output of from a real QRNG see, for example, Ref. [21].

4.3 Discussions

The entropy for the unbiased random numbers obtained from 10 GBit raw data is 7.999999 per byte and is consistent with the ideal value of 8. The data passes all tests in standard NIST and diehard statistical test suites. Moreover, in Ref. [22] the quantum random bits were also analyzed with a test more directly related to the algorithmic randomness of a sequence (rather than simply statistical properties). Specifically, the raw bits were used to test the primality of all Carmichael numbers smaller than 54×107 with the Solovay-Strassen probabilistic algorithm, and the minimum random bits necessary to confirm compositeness was used as the metric. Ten sequences of raw quantum random bits of length 229 were compared with sequences of the same length from three modern pseudo-random generators (Random123, PCG and xoroshilro128+) and some advantage was found using the quantum bits [13]. In summary, we experimentally realized the QRNG certified with the Kochen-Specker the-

78 4.3. DISCUSSIONS orem. While our QRNG is not certified in device independent the certification scheme allows to establish a low bound on value-indefiniteness of each output of the QRNG in contrast to the certification schemes based on the Bell-type inequalities which only provided the lower bound on the generating process entropy but did not give any predictions for a specific output. On a more practical level our QRNG eliminates the necessity for input seed random numbers, lifts the non-locality requirements for the certified generator which, in turn, greatly enhances the rate of generation of certified random numbers. The rate of generation of 25 kBit/s of unbiased random bit is limited by the qutrit decay rate (T1 ∼ 5 µs) and may be further increased by using active schemes for initialization of the system in the ground state [23, 24]. The certification confidence of 99.7% can be improved by using qutrits with longer relaxation times or applying more post-processing to the raw generated data. Additional tests on the outcomes of our QRNG reinforce our conclusion that the QRNG is ready for use in real-life applications given that the confidence bound on randomness is tolerated.

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[18] M. Jerger, P. Macha, A. R. Hamann, Y. Reshitnyk, K. Juliusson, and A. Fedorov, Phys. Rev. Appl. 6, 014014 (2016).

[19] C. Eichler, Y. Salathe, J. Mlynek, S. Schmidt, and A. Wallraﬀ, Phys. Rev. Lett. , 110502 (2014).

[20] A. A. Abbott, C. S. Calude, and K. Svozil, Journal of Mathematical Physics 56, 102201 (2015).

[21] D. Frauchiger, R. Renner, and M. Troyer, arXiv:1311.4547 (2012).

[22] A. A. Abbott, C. S. Calude, M. J. Dinneen, and N. Huang, “Experimental evidence of the superiority of quantum randomness over pseudo-randomness,” (2017).

[23] K. Geerlings, Z. Leghtas, I. M. Pop, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, arXiv:1211.0491 (2012).

[24] D. Rist`e,J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo, Phys. Rev. Lett. 109, 050507 (2012).

81 BIBLIOGRAPHY

82 CHAPTER 5

IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

Characterization of the frequency response of an electronic component or a control line is a routine procedure in virtually any experimental lab. However, low-temperature research created new challenges for these types of measurements because conventional electronic devices used for this purpose are not suitable: first, neither oscilloscopes nor vector network analyzers work at cryogenic temperatures and, second, they cannot be directly connected to a component on a chip. In addition, the quantum control experiments and quantum information processing set the strictest requirements on the fidelities of quantum operations and that, in turn, requires precise calibration of the control lines. This problem has been widely recognized by the quantum information community and it has been accepted that the ultimate solution to it for experiments with qubit control is to use the qubit itself as an in-situ probe of the signal. The first attempts to do so were performed as early as in 2009 [1]. This topic has the primary focus of Refs. [2,4] and was addressed in many other works (see, for example, Refs. [3,5], or the recent paper Ref. [6]). Yet, no practical methods for direct characterization of a control line have been developed up to date. In this chapter we extend the discussion of in-situ quantum control and present such a method. More specifically, we propose and experimentally realize a technique to measure the transfer function of a control line in the frequency domain using a qubit as a vector network analyzer. Our method requires coupling the line under test to the longitudinal component of the Hamiltonian of the qubit and the ability to induce Rabi oscillations through simultaneous driving of the transverse component. Following a brief introduction, we describe the theoretical idea underlying our method and extend the theoretical description to a non-resonant driving case allowing us to extend the frequency range of our method beyond any of the existing ones in the high-frequency part of the spectrum. We then move on to describe the experimental procedure and give technical details useful for its efficient implementation. We demonstrate the applications of the method by characterizing the ’flux’ control line of a Transmon qubit in the range from 1 to 450 MHz and using this characterization to improve the fidelity of an entangling CPHASE gate between two qubits. Finally, we discuss the fundamental accuracy limits of the method and present the analysis of its errors performed by the means

83 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER of numerical simulations. The results of this chapter are published in Phys. Rev. Lett. 123, 150501 (2019) [21].

5.1 Introduction

Signal distortions are inevitable in experiments involving radio frequency controls, where they can impact the quality of measurements and generate unwanted artifacts. In quantum control experiments and quantum information processing, these distortions are the source of errors and may limit the fidelity of operations. Quantum gates that use nonadiabatic (fast) frequency tuning of the qubits involved are particularly sensitive to distortion and require precise calibration [1–6]. Distortion can be canceled, in principle, if the complex transfer function of the control line is known, by applying its inverse to the signal before it is transmitted. The most common approach to obtain the transfer function is to measure it (at room temperature) in the frequency domain using a vector network analyzer or in the time domain using an oscilloscope (see, for example, Ref. [5]). This method has two important deficiencies: the transfer function of the line changes when the setup is cooled to cryogenic temperatures, and the part of the signal line from the microwave connector closest to the chip to the qubit is not included in the characterization. Various methods for in situ line calibration have been proposed. Some calibration methods are limited in time resolution by the length of the microwave π-pulse [1,3,5], others are applicable only to specific systems [2] or pulses [7], and most procedures only provide indirect information about the transfer function. The main problem of characterization of the frequency (z) control line with the qubit in the frequency domain is that the modulation of the qubit frequency does not change the excited state population of the qubit, the quantity which is normally accessible for detection in an experiment. To go around this problem we propose a simple but elegant idea to generate Rabi oscillations of the dressed-state qubit following the observation that in the rotating frame the frequency control does not commute with the qubit Hamiltonian and generates excitations. We demonstrate validity and practicality of our method by directly measuring response of the z-control line in a wide-range of frequencies which has never been done before. Our method can be immediately used to increase fidelities of two-qubit quantum gates for superconducting qubits in the most popular state-of-art implementations of quantum algorithms (for example, ones which use flux tuning for the implementation of the two-qubit gates which were originally proposed in Ref. [7,8]). More generally our method can provide significant improvement for any algorithm involving non-adiabatic control of the qubit frequency (Refs. [4, 6]). In this Chapter we propose and experimentally realize a method of in situ direct reconstruction of the response of a control line of a qubit using the qubit itself. The accuracy of our method is limited by decoherence of the qubit at low frequencies. At high frequencies the accuracy is fundamentally limited by the rotating wave approximation (see Fundamental accuracy

84 5.2. THEORY

Figure 5.1: (Top) Prepared qubit pulse is being distorted by the control line. Resulting signal is the convolution (in time domain) of the programmed pulse and the impulse response of the control line. (Bottom) Knowing the transfer function G(ω) or an impulse response g(t) allows applying its inverse to the input in pre-processing and feeding the resulting pulse through the control line.

limits). We benchmark the method by measuring the transfer function of an element introduced at room temperature and comparing the measured response with the response obtained by using a commercial vector network analyzer (VNA). We then apply the method to improve the ﬁdelity of a non-adiabatic controlled phase (CPHASE) gate [9, 10] between two Transmon qubits, one of the most commonly used entangling gates in superconducting systems.

5.2 Theory

To understand the principles underlying our method consider the Hamiltonian of a qubit with time-dependent longitudinal (frequency control) and transverse (excitation) drives

h̵ H = ω σ + hA̵ cos (ω t + φ ) σ 2 q z x x x x ̵ + hAz cos (ωzt + φz) σz. (5.1)

85 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

Ignoring the longitudinal drive for now, one might recall that resonant driving with the drive frequency ωx being equal to the qubit level splitting ωq leads to Rabi oscillations, which can easily be demonstrated by switching to the frame rotating with the frequency of the drive:

H 1 = ω σ + A cos (ω t + ϕ ) σ h̵ 2 q z x x x x i S U = exp ω tσ ω = ω (5.2) 1 2 x z x q H′ 1 = A cos ϕ σ′ + sin ϕ σ′ h̵ 2 x x x x y

Figure 5.2: Left: Rabi oscillations in the ﬁrst rotating frame. Right: Eﬀective Rabi oscillations of a dressed qubit in the second rotating frame.

Here we have used the rotating wave approximation under the assumption Ax ≪ ωq and omitted the fast oscillating terms proportional to cos (2ωxt). Note that while the phase of the drive explicitly appear in the expression here, it is actually lost in the experiment unless another pulse with a diﬀerent phase is also used. An example would be a tomography pulse used to reconstruct the qubit state after performing the experiment, which would serve as a phase reference.

Recalling the longitudinal (σz) qubit control to AC-drive the qubit rather than shift its energy splitting for a ﬁxed period of time, we can write a Hamiltonian of the qubit in the ﬁrst rotating frame (taking the phase ϕx to be zero):

H′ 1 = A σ′ + A cos (ω t + ϕ ) σ′ (5.3) h̵ 2 x x z z z z

Note again, that the σz driving term is intact by the first rotating frame transformation as it commutes with the rotation unitary operator. Comparing (5.2) and (5.3), we observe that in the rotating frame the z term plays the role of a transverse drive for the dressed-state qubit with splitting Ax and will induce Rabi oscillations of the dressed-state qubit with frequency Az. In the same fashion as inducing Rabi oscillations for the Hamiltonian 5.2 required fulfilling the resonant condition ωx = ωq (charge drive is on resonance with the qubit energy splitting), here one must ensure driving the flux (longitudinal coupling) line on resonance with the dressed qubit’s effective energy splitting: ωz = Ax. To make this observation explicit we transform the Hamiltonian5 .3 into the second rotating frame

86 5.2. THEORY

i(ω t)σ˜′ ~2 for ωz = Ax with the unitary of e z x to obtain

H′′ 1 = A sin φ σ′′ + cos φ σ′′ , (5.4) h̵ 2 z z y z z where we have used another rotating wave approximation with Az ≪ Ax. The Hamiltonian ′′ H shows that the amplitude Az and phase φz of the z control at the driving frequency ωz are encoded in the frequency and axis of the Rabi oscillations of the dressed-state qubit and can be measured in the experiment.

5.2.1 Non-resonant driving case

Consider a general transformation of the Hamiltonian (5.1) into the rotating frame with

i(ωxt+φx)σz~2 U1 = e ,

but now allowing the drive frequency ωx to be diﬀerent from the qubit splitting ωq. The Hamiltonian reads h̵ h̵ H′ = A σ′ + δωσ′ + hA̵ cos (ω t + φ ) σ′ , (5.5) 2 x x 2 z z z z z where δω = ωq − ωx is the detuning of the excitation driving and we have used the rotating wave approximation under the assumption Ax ≪ ωq. Note again that the transverse phase φx does not explicitly appear in the Hamiltonian (5.5), however it implicitly provides a reference for the rotating frame through the transformation operator U1. As the Hamiltonian (5.5) has ′ an extra ’driving’ term proportional to σz compared to (5.3), one needs to do an intermediate step and diagonalize the time-independent part of H′ in (5.5). ′ iφtσ ~2 It can be done by applying a time-independent unitary U2 = e y , with φt = arctan (δω~Ax), which leads to

h̵ H˜ ′ = Ω σ˜′ + hω̵ cos (ω t + φ ) σ˜′ 2 R x R z z z ̵ δω ′ + hωR cos (ωzt + φz) σ˜x, (5.6) Ax » 2 2 where ΩR = Ax + δω is the full Rabi frequency and ωR = AzAx~ΩR. Comparing this expression to (5.3) gives us an important observation: the eﬀective Rabi frequency of the dressed qubit is now higher (ΩR instead of just Ax) and can be increased by ′ driving the qubit with more detuning. At the same time, the oscillatingσ ˜x term does not substaintially aﬀect the dynamics of the systems and can be omitted. Worth checking that in the special case of δω = 0, U2 reduces to the identity, ΩR = Ax and ωR = Az, and the expressions (5.6) and (5.3) coincide.

Similarly to the resonant case, we can consider it a dressed-state qubit with splitting ΩR ′ driven by the transverse drive (˜σz term). As the main reason to consider the oﬀ-resonant driving in the lab frame is to achieve the increase in the Rabi frequency (see below in the text), we are not going to consider the oﬀ-resonant driving of the dressed qubit in the rotating frame

87 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

and would still fullﬁll the resonant condition ωz = ΩR. Analogously to the resonant driving case, it will induce Rabi oscillations of the dressed-state qubit with frequency ωR. To make this observation explicit we transform the Hamiltonian (5.6) into the second rotating frame for i(ω t)σ˜′ ~2 ωz = ΩR with U3 = e z x to obtain

h̵ H′′ = ω sin φ σ′′ + cos φ σ′′ , (5.7) 2 R z y z z where we have used another rotating wave approximation with ωR ≪ ΩR. In full analogy to ′′ (5.4), the Hamiltonian H in (5.7) shows the encoding of the amplitude Az (via ωR) and phase

φz of the z control in the frequency and axis of the Rabi oscillations of the dressed-state qubit and allows them to be measured in the experiment.

5.3 Experimental procedure

The experimental procedure is summarized by the following steps.

• Apply x drive with z drive oﬀ (Az = 0) and ﬁt data to extract ΩR (Fig. 5.3a).

′ • Set ωz = ΩR, apply both x and z drives and use tomography pulses to reconstruct ⟨σx(t)⟩, ′ ′ ⟨σy(t)⟩ and ⟨σz(t)⟩ in the ﬁrst rotating frame (Fig. 5.3b).

′′ ′′ ′′ • Post-process the data to reconstruct ⟨σx (t)⟩, ⟨σy (t)⟩ and ⟨σz (t)⟩ in the second rotating

frame. Fit the resulting data to Rabi oscillations described by (5.7) to extract Az and φz

for given ωz (Fig. 5.3c).

• Repeat the sequence for diﬀerent ΩR to cover the necessary frequency range.

Both Az and phase φz for a given frequency ωz can be determined directly from the observables in the laboratory or the ﬁrst rotating frames, such as the excited state population of the qubit. However, it is more convenient to perform tomography and reconstruct the oscillation of the qubit state in the second rotating frame by post-processing. Removing the fast population oscillation with frequency ΩR and leaving only the signal varying with the frequency ωR allows for a substantial reduction of the required sampling rate and more robust ﬁtting.

By setting δω = 0 and varying Ax one can perform the experiment for different ωz and identify the transfer function of the z line in the range of frequencies Γ1, Γ2 ≪ ωz ≪ ωq which are most relevant for the frequency control of the qubit with Γ1,2 being the relaxation and dephasing rates of the qubit, respectively. The idea of scanning the Rabi frequency was also used to perform noise spectroscopy [14, 15] utilizing the dependence of the Rabi decay time on the Rabi frequency [16]. For highly anharmonic flux qubits Rabi frequencies of up to 1.7 GHz were achieved [15]. For weakly anharmonic qubits, such as the Transmon, the two-level approximation breaks down when Ax starts being a considerable fraction of the anharmonicity, about 350 MHz for the qubits used here. To mitigate this problem, we can exploit off-resonant driving (δω < 0),

88 5.4. DETAILED EXPERIMENTAL PROCEDURE

(a) |0 (b) |0 (c) |0

y y y

x x x

|1 |1 |1

1 1 1 x x x 0 0 0 0 0 0 0

-1 -1 -1 1 1 1 y y y 0 0 0 0 0 0 0

-1 -1 -1 1 1 1 z z z 0 0 0 0 0 0 0

-1 -1 -1 0 250 500 750 1000 0 250 500 750 1000 0 250 500 750 1000 Pulse Duration (ns)

Figure 5.3: A typical data set for ΩR = 19.8 MHz and the corresponding trajectories of the Bloch vector in the first rotating frame with only x drive applied, with both x and z drives and with both x and z drives in the second rotating frame. a) State tomography of the qubit with only the x drive applied. ⃗ The fit yields a rotation vector of Θ=(19.7,-1.9,0.8) MHz and φx = −0.1. b) State tomography of the qubit with both the x and z drives applied. The frequency of z drive is set to ωz = ΩR. The sampling rate of the experiment is chosen such that the slow oscillations at ωR can be resolved unambiguously but not necessarily the fast oscillations at ΩR, which are canceled by U3. The orange curve shows the theoretical dynamics derived from the fit in (c). c) The dynamics of the qubit in the second rotating ⃗ frame, transformed from (b). The fit yields θ=(0.2,-3.5,0.1) MHz and we can obtain Az = 3.51 MHz and φz = −1.54.

which increases the ΩR at the same Ax and also allows higher Ax because the frequency of the drive is further from the frequency of the S1⟩-S2⟩ transition. This allows us to extend the analysis to higher frequencies at the expense of signal amplitude.

5.4 Detailed experimental procedure

In this section we go through the experiment step-by-step, giving some directions for efficient implementation of the procedure. We also discuss some of the complications we have encoun- tered performing the measurements and ways to mitigate them. Step 0: Make sure Eq. (5.1) is a good approximation of the physical system. In case of the Transmon and other superconducting qubits, that means choosing a magnetic flux bias and offset (longitudinal driving) amplitude that result in a predominantly linear dependence of the qubit transition frequency on the offset. Choosing a bias point too close to the symmetry point of the qubit would result in a non-

89 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER linear dependence of the qubit energy splitting on flux even for weak longitudinal driving. In ̵ other words, the term hAz cos (ωzt + φz) σz in (5.1) will have to be modified to account for this nonlinearity resulting in a complicated dynamics. In order to avoid that, the Az amplitude has to be small compared to the detuning of the qubit from its symmetry point. Step 1: Apply an x drive to the qubit with the z drive off. The x drive can use square pulses or any pulse shape with an envelope that has a constant amplitude section. Sweep the duration of the constant amplitude section to observe Rabi oscillations. Perform tomography ′ ′ ′ to reconstruct the evolution of the state vector v⃗(t) = ⟨σx(t)⟩, ⟨σy(t)⟩, ⟨σz(t)⟩ in the first rotating frame (see Fig. 5.3a). ⃗ ⃗ ⃗ Fit exp(ΘRt · L)v⃗(0) to this data with fitting parameters ΘR = (Θx, Θy, Θz). Here, t is the ⃗ 0 0 0 duration of the constant amplitude section of the pulses and L = (Lx,Ly,Lz), Lx = 0 0 −1 , 0 1 0 0 0 1 0 −1 0 Ly = 0 0 0 , Lz = 1 0 0 are the SO(3) generators of rotations about the x, y and z axis, −1 0 0 0 0 0 respectively, and v⃗(0) = (0, 0, −1) is the initial state of the system. The direction of Θ⃗ determines the axis of rotation of the state vector on the Bloch sphere and the absolute value gives the ⃗ Rabi frequency SΘS = ΩR. If non-square pulses are used, v⃗(0) can be made a fitting parameter to absorb the attack of the pulses. ⃗ Ideally we expect Θ = (Ax, 0, δω). However in the experiment we often observed spontaneous detuning of the qubit Θz ≠ 0 at larger amplitudes Ax due to the presence of higher energy levels in the transmon (see error analysis below). As an example, the data for ΩR = 19.8 MHz was ′ taken assuming δω = 0 but shows (see Fig. 5.3a) small oscillations for ⟨σx(t)⟩ component.

In addition, we also observed a spurious component Θy ≠ 0 due to phase differences between the tomography pulses and the qubit driving pulse. In particular, for δω ≠ 0 the frequency of the driving pulse is different from the frequency of the resonant tomography pulses which may lead to some phase difference depending on the transfer function of the x (transverse) drive line and DACs. This effect can be canceled in the post-processing of the data in Step 3. ⃗ Step 2: Apply x-drive as in Step 1, add simultaneous z-drive at frequency ωz = SΘS. Use a square pulse envelope for the z driving pulses and sweep the duration of the x and z pulses. ′ ′ ′ Use tomography to reconstruct the evolution of a state vector v⃗0(t) = ⟨σx(t)⟩, ⟨σy(t)⟩, ⟨σz(t)⟩. Due to x-drive the y and z components of the state vector oscillate with the Rabi frequency

ΩR while the z-drive modulates these oscillations at ωR ≪ ΩR (see Fig. 5.3b). The duration of the pulses should be long enough to observe this slower modulation. Step 3: Post-process the data to reconstruct the state vector rotation in the second rotating frame. That step has the benefit of removing the fast Rabi oscillations, thus allowing to take fewer measurements, and allows to fit simpler expressions with less parameters to the data. To post-process the data we first correct for the phase shift between the x driving pulse and tomography pulses by applying the transformation v⃗1(t) = exp(φxLz)v⃗0(t), where φx = arctan (Θy~Θx). To account for the transformation from (2) to (3) in the main text we rotate 1~2 2 2 the basis v⃗2(t) = exp(φdLy)v⃗1(t), where φd = − arctan Θz~ Θx + Θy . Finally we remove ⃗ the fast oscillations by going to the second rotating frame v⃗3(t) = exp(SΘSLxt)v⃗2(t). The resulting Rabi oscillations of the state vector in the second rotating frame described

90 5.5. REALIZATION

100

z 80 A

d 60 u t i l 40 p (MHz/V)

m 20 A

0.2

0.0 z

-0.2 e s

a -0.4 h P -0.6

-0.8 0 100 200 300 400 500 Frequency (MHz)

Figure 5.4: Amplitudes Az and phases φz of the transfer functions of the z control line. Points in blue and green show the response of the flux line for two different configurations (see text for details). Darker points were measured with resonant driving ωx = ωq by varying Ax, lighter points were measured by varying ωx − ωq with fixed Ax (off-resonant case).

′ ⃗ ⃗ by v⃗3(t) = (⟨σ”x(t)⟩, ⟨σ”y(t)⟩, ⟨σ”z(t)⟩) (see Fig. 5.3c) are ﬁt to exp(θR · L)v⃗(0), where both ⃗ θ = (θx, θy, θz) and v⃗(0) are ﬁt parameters. The amplitude of the z-drive can be found as 1~2 ⃗ ⃗ 2 2 Az = SθSSΘS~ Θx + Θy and the phase of the drive is given by φz = arctan(θy~θz). Step 3 completes the characterization of the z-line at the frequency ωz = ΩR and the procedure can be repeated for another frequency ΩR controlled by the appropriate choice of Ax and ωx.

5.5 Realization

We have tested our method on a standard circuit quantum electrodynamics system: a Transmon qubit coupled to a readout resonator with local charge and ﬂux lines. More speciﬁcally, we choose the QB 2 of a chip virtually identical to one used in Ref. [8]. With our method we characterized the complex transfer function in the range of 1 to 450 MHz (Fig. 5.4). Each point was taken with 4,096 averages at a repetition rate of 40 kHz. The measurements at lower frequencies were limited by decoherence of the qubit. However, we point out that the low frequency part of the transfer function can be also measured with other

91 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

100

10 1 Amplitude

10 2

1.0

0.5

0.0

Phase -0.5

-1.0

60 80 100 120 Frequency (MHz)

Figure 5.5: Amplitude and phase of transmission through a transmission line with a shorted stub resonator. Points in blue were measured with the qubit by comparing Az and φz with and without the stub in place, the orange line was measured directly with a commercial vector network analyzer.

methods with lower time resolution [1,3]. At high frequencies our accuracy is limited by population of the higher levels for the resonant driving case and by the loss of signal contrast at large detunings in the oﬀ-resonant driving case. In addition, the amplitude reconstruction is more robust as it corresponds to the frequency of the oscillations while the phase is reconstructed from a ratio of their amplitudes. We have benchmarked our method by introducing an additional element in the ﬂux line at the room temperature. We used a shorted stub resonator made from a BNC T-adapter and several meters of a BNC cable shorted at the end. We repeated the characterization of the line with the element and used the original data for the line to de-embed the transfer function of the element itself. The result is shown in Fig. 5.5 in comparison with the transfer function of the element measured separately with the commercial vector network analyzer (VNA). The agreement between our method and VNA is excellent, showing the dynamic range of our method for measuring amplitude of ≃ 30 dB. The dynamic range of a single measurement is bounded by the condition ωR ≪ ΩR required for the rotating wave approximation in (5.7) and the decoherence time of the qubit. It can be further improved by dynamically changing the amplitude of the z drive, taking advantage of the dynamic range of the AWG.

92 5.5. REALIZATION

100 (a) (b) (c)

40 Pulse length (ns)

0 0.9 1.0 1.1 0.9 1.0 1.1 0.9 1.0 1.1 Pulse amplitude

Figure 5.6: Vaccum Rabi oscillation between the S11⟩ and S20⟩ states of two Transmon qubits assuming (a) a perfect impulse response, (b) the impulse response measured at room temperature and (c) the impulse response measured using the qubit.

We have employed the outlined method of flux line calibration to improve the quality of a CPHASE entangling gate between two superconducting transmon qubits. The gate between two transmons can be realized [9, 10] by bringing the second excited state S02⟩ of one qubit in resonance with the S11⟩ state where both qubits are excited, inducing energy exchange between the states. On each swap of an excitation the state acquires a phase of π~2, so that a system starting in S11⟩ evolves into i S02⟩ and subsequently into − S11⟩. Since one of the qubits must be tuned in frequency, there is an additional dynamical phase acquired by the qubit which must be accounted for. Apart from that, the other computational basis states, S00⟩, S10⟩ and S01⟩ are unaffected by the gate. To implement the gate we used QB1 and QB2 of the chip both parked at their symmetry points at frequencies 5.5 GHz and 6.5 GHz, respectively. We tune QB2 near ∼ 5.8 GHz to allow the states S11⟩ and S02⟩ to exchange excitation via virtual photons in a common transmission line resonator. An ideal CPHASE gate requires a perfect step pulse, which cannot be implemented by any physical device. In order to comply with bandwidth limitation of our AWG and to make the distortion correction easier, we applied a Gaussian low-pass filter with a cut-off frequency of

93 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

300 MHz to the ideal square pulse shape. This pulse shape was programmed to the AWG and was used to implement the CHPASE gate. To calibrate the amplitude and duration of the flux pulse we prepared qubits in S11⟩ followed by the flux pulse on QB2 where we sweep both the length and the amplitude of the pulse. Measurement of the resonator response shows characteristic oscillations (often called ’chevron patterns’) manifesting excitation exchange between S11⟩ and S02⟩ states of the two transmons. The pattern is distorted due to extra reflections on the flux line which appear only at low temperatures and cannot be calibrated at room temperatures (the reflections of the line can be seen as broad resonances in the line frequency response, see Fig. 5.6a). Following the most common approach to compensate for these distortions, we have measured the room temperature (RT) impulse response of the flux line before cooling down the refrigerator. We then used the standard machinery [5] to compute the necessary waveform which then can be loaded to AWG to yield the desired pulse shape. Using this computed pulse shape, we repeated the calibration routine (Fig. 5.6b) but the oscillations remain heavily distorted. In particular, one may note that the oscillations start only after the first fifteen nanoseconds, which we attribute to a large overshoot in the actual flux pulse. With our in situ calibration of the flux line (the response in blue on Fig. 5.4) for distortion compensation, we have measured the improved oscillation pattern shown in Fig. 5.6c. The oscillations visibility is asymmetric which we attribute mostly to the bandwidth limitations of the pulse shape. Apart from that the compensation fixes most of the imperfections and more specifically the oscillations begin without delay at short lengths of the flux pulse. To quantify our method, we have calibrated the CPHASE gate with no correction, with RT correction and our in situ correction. Applying our method the process fidelity of the gate increases from 0.835±0.015 to 0.875±0.01. Numerical simulation of the protocol with the experimental values of T1 = 2.5 µs and T2 = 1.4 µs and perfect pulse shape showed the process fidelity of 0.885. Contrary to our initial expectations, applying the RT calibration has not improved the fidelity of the gate. During a later cooldown we introduced extra attenuation to the resonator input and flux lines which increased the coherence of the qubits to T1 = 4 µs, T2 = 4.5 µs and reassembled the flux control line. In this configuration we were unable to demonstrate a statistically significant improvement in the fidelity of the CPHASE gate, neither with RT-calibration nor with in situ calibration. The process fidelity was measured to be 0.945, which was predominately limited by decoherence of the transmons. In situ measurement of the frequency response of the line (Fig. 5.4, green curves, rescaled for the extra attenuation) did not show the characteristic resonances seen previously (Fig. 5.4, blue curves). Both measurements reinforce our statement that using the qubit as a VNA allows one to precisely characterize the qubit control lines in a methodical manner and improve the fidelity of entangling gates if the control signals’ distortions contribute significantly to the infidelity. Our method is the first direct in situ measurement of the line transfer function from room temperature electronics to a qubit on a chip. The method is most relevant for superconducting

94 5.6. QUALITY OF THE PHASE RECONSTRUCTION

y 20 z

0 Amplitude (MHz/V) 20 g r p z 2.61 2.60

z 2.59 0 /2 3 /2 2 prg Programmed z

Figure 5.7: Quality of the phase reconstruction – (top) Measured y and z components of the Rotation vector of the dressed qubit as the phase of the longitudinal driving pulse programmed at the AWG is varied. (bottom) Diﬀerence between the phase φz extracted by the method and the phase programmed at the AWG.

qubits whose frequencies are routinely tuned but is applicable for all qubits with z and x control. For superconducting qubits one can use our procedure to improve the ﬁdelity of the two-qubit quantum gates [9–12] as well as photon-qubit operations requiring non-adiabatic control [1,4,6]. In addition to quantum control applications the qubit can be also used as a microscopic probe of the electromagnetic ﬁelds in frequency domain.

5.6 Quality of the phase reconstruction

The sensitivity of our method to the phase of z-drive is demonstrated in Fig. 5.7. As expected a change in phase of the z-driving is reliably detected by our procedure. This should be contrasted to the conventional Rabi oscillations as only the amplitude of the drive can be accessed from the frequency of the oscillations while the phase is not accessible.

95 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

5.7 Reconstruction of the transfer function of the ’charge’ (transverse) control line

Our method can be simplified to measure the transfer function of the ’charge’ (transverse) control line of a qubit around the qubit transition frequency – i.e., in the range where the correction to the control signals may be required. In order to measure the amplitude Ax of the transfer function it is enough to perform a standard Rabi type experiment with detuned driving. This experiment also serves as the first part of the longitudinal drive line calibration method (the first bullet point or Step 1 in Experimental procedure), but here we vary the drive detuning instead of driving amplitude. Performing state tomography on the result allows » 2 2 determining the Ax at different detunings; the full measured Rabi frequency is Ax + δω .

In order to reconstruct the phase φx of the transfer function no extra measurements are required. Tomography pulses are always delivered at the qubit frequency and therefore provide a fixed phase reference. The phase of Rabi oscillations (axis on the Bloch sphere), however, depends on φx and allows to reconstruct it. Step 1: Apply an x drive to the qubit with the z drive off. The x drive can use square pulses or any pulse shape with an envelope that has a constant amplitude section. Sweep the duration of the constant amplitude section to observe Rabi oscillations. Perform tomography to ′ ′ ′ reconstruct the evolution of a state vector v⃗(t) = ⟨σx(t)⟩, ⟨σy(t)⟩, ⟨σz(t)⟩ in the first rotating frame (see Fig. 5.3a). ⃗ ⃗ ⃗ Step 2: Fit exp(ΘRt · L)v⃗(0) to this data with fitting parameters ΘR = (Θx, Θy, Θz). Here, ⃗ t is the duration of the constant amplitude section of the pulses and L = (Lx,Ly,Lz), Lx = 0 0 0 0 0 1 0 −1 0 0 0 −1 , Ly = 0 0 0 , Lz = 1 0 0 are the SO(3) generators of rotations about the x, y and 0 1 0 −1 0 0 0 0 0 z axis, respectively, and v⃗(0) = (0, 0, −1) is the initial state of the system. The direction of Θ⃗ determines the axis of rotation of the state vector on the Bloch sphere and the absolute value ⃗ gives the Rabi frequency SΘS = ΩR. If non-square pulses are used, v⃗(0) can be made a fitting parameter to absorb the attack of the pulses. ⃗ Step 3: Ideally we expect Θ = (Ax, 0, δω). In the case of the transverse drive transfer function measurement there is no requirement to go higher in Rabi frequency, so we can avoid spontaneous detuning at larger amplitudes Ax. However, as we vary δω, we see the additional component Θy ≠ 0 due to phase differences between the tomography pulses and the qubit driving pulse. Those phase differences arise from the transfer function φx of the control line and DACs. Taking into account the former, we obtain the phase part of the transfer function.

5.8 Fundamental accuracy limits

Our method is based on the ability to generate Rabi oscillations and extraction of the frequency and amplitude of these oscillations in the ﬁrst and second rotating frames. Thus, the most fundamental limits for our method are errors due to the rotating wave approximation (RWA), which deﬂect the state vector dynamics from an ideal oscillatory behaviour. Let us consider

96 5.8. FUNDAMENTAL ACCURACY LIMITS these errors in more detail.

5.8.1 Rabi frequency change due to the RW approximation

To estimate the error due to the RW approximation, consider a general model of a qubit driven resonantly by an external transverse ﬁeld:

H ω Ω = qb σ + R σ cos (ω t) (5.8) h̵ 2 z 2 x qb

When the driving strength increases, the trajectory of a qubits state on a Bloch sphere deviates from simple Rabi oscillations and includes cycloidal-like motions known as Bloch- Siegert oscillations, whose period is equal to the drive period [18, 19]. An eﬀective rotating- frame Hamiltonian describing the evolution of the qubit (corrected for the fast oscillations) is given by

Heﬀ ΩR ΩR ΩR ̵ = σx − σz (5.9) h 2 2 8ωqb 2 3 ΩR ΩR ΩR − σx + O 2 8ωqb ωqb

Here, the first term describes the ideal Rabi oscillations, the second term is known as the Bloch-Siegert shift and the third terms results in decreasing the effective Rabi frequency. The Bloch-Siegert shift will tilt the rotation axis of the state vector on the Bloch sphere but will only contribute to the effective Rabi frequency in the second order. Overall, the effective Rabi frequency up to the second order in ΩR~ωqb is given by

⎛ 2⎞ eﬀ 1 ΩR ΩR = ΩR 1 − (5.10) ⎝ 2 8ωqb ⎠

5.8.2 Error of reconstruction in the ﬁrst rotating frame

In the ﬁrst rotating frame the Rabi frequency will be interpreted as the frequency which we probe the line response at. This change of the eﬀective Rabi frequency will then contribute to an error of reconstruction of Az as

2 δAz(ω) 1 ω SδAz(ω)S = (5.11) δω 2 8ωqb

δAz(ω) As an example, for an ideal two-level system with ωqb = 3 GHz and δω ∼ 1 we get

SδAz(ω)Sω=300 MHz ∼ 0.1%. For a transmon qubit, however, the error will be fully dominated by the excitation of higher levels rather than by the error of the RWA. Using our oﬀ-resonant method we can decrease the error due to higher level excitation and, at least in principle,

97 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER reach the fundamental limit of the RW approximation albeit at the expense of lowering the signal-to-noise ratio and long measurement times.

5.8.3 Error of reconstruction in the second rotating frame

In the second rotating frame ΩR will play a role of a dressed qubit and the dressed qubit Rabi frequency ωR is given ideally by the amplitude of the longitudinal driving Az = ωR. The change in the eﬀective Rabi frequency will then directly contribute to the error of the Az reconstruction as

2 (1) ωR ωR UδAz (ω)U = (5.12) 2 8ω

To keep this error small it is preferable to keep ωR as low as possible. However, ωR is limited from below by decoherence of the qubit. The effective Rabi frequency ω eff recovered from a » R eff 2 2 fit to damped oscillations is given by ωR = ωR − Γ , where Γ ≃ max (Γ1, Γ2)) is the decay rate of the Rabi oscillations. The error of the amplitude reconstruction due to damping is then given by 2 (2) ωR Γ UδAz (ω)U = (5.13) 2 ωR √ (1) (2) 2 An optimal ωR will be achieved when UδAz (ω)U+UδAz (ω)U is minimum, at ωR = (8~ 3) Γω 1~2 −(1~2) −(3~4) Γ with minimal error SδAz(ω)S = 2 3 Γ ω where we used Γ ≪ ω. This result shows Γ 1~2 that the error at low frequencies will increase as 2ω and will ultimately be limited by the decoherence rate of the qubit. Summary: The fundamental accuracy of our method is limited by the RWA for high frequencies. For low frequencies the limit is set by the trade-off between the RWA error in the second rotating wave and decoherence of the qubit.

5.9 Error analysis

In this section we discuss the error budget of the method and provide error estimates for the transfer function reconstruction. We also suggest pathways to decrease some of the errors which come from the reconstruction procedure and do not depend on the limitations of the model. Apart from statistical errors, which can be addressed by increasing the number of averages (or using quantum limited ampliﬁers) and will not be thoroughly discussed here, several sources of systematic errors contribute to incorrect transfer function reconstruction. Among these errors are population of the higher levels of the transmon – i.e., the breakdown of the qubit approximation; decoherence of the qubit and errors due to two rotating wave approximations. To estimate the errors we performed a full numerical simulation of a driven transmon circuit including decoherence using the qutip package [20] and truncating our Hilbert space to ten lowest energy eigenstates of the transmon. First, consider Step 1 of the experiment where the qubit is driven on resonance by an x drive

98 5.9. ERROR ANALYSIS

x 0.3 y z

R 0.2 qubit R

0.1

0.0 Fitted frequency, GHz 0.1

0.0 0.1 0.2 0.3 Input Rabi frequency Ax, GHz ⃗ Figure 5.8: Amplitude Ax and rotation vector Θ reconstruction – Rotation of the bloch vector of a transmon circuit compared to a two-level system. With increasing driving amplitudes, the Rabi frequency of the transmon is decreased compared to the qubit and a spontaneous detuning can be observed.

with the z drive off. As the Rabi amplitude Ax increases and approaches the anharmonicity, the higher levels of the transmon are populated and the two-level approximation breaks down. In particular, already at relatively low amplitudes ≃ 0.1 GHz we observe spontaneous detuning of the qubit from its bare frequency (see Fig. 5.8). This problem can be mitigated by taking this detuning in into account as an extra parameter in the analysis. At larger Ax the effects due to population of higher levels cannot be reduced to a simple correction of the Rabi rotation axis and is, therefore, introducing an error to both phase and amplitude reconstruction in the following steps of the experiment. Figures 5.9 and 5.10 show the difference between the phase and amplitude set in the Hamiltonian and the phase and amplitude reconstructed from the simulated data following the procedure identical to the experimental one. This simulation shows that the effects of decay, detuning and counter-rotating terms can be fully included in the model and yield no systematic error. In agreement with simulation of the Rabi experiment our results show that the major systematic error of the method comes from the two-level approximation of the transmon circuit which grows with frequency. As expected, off-resonant driving of the qubit allows to extend the

99 CHAPTER 5. IN SITU CHARACTERIZATION OF QUBIT CONTROL LINES: A QUBIT AS A VECTOR NETWORK ANALYZER

0.30 Transmon resonant sweep Transmon off-resonant sweep 0.25 qubit resonant sweep qubit off-resonant sweep

z 0.20 A

n i

r 0.15 o r r e 0.10 e v i t a l 0.05 e R 0.00

0.05

0.10 0.0 0.1 0.2 0.3 0.4 0.5 Fitted Rabi frequency R, GHz

Figure 5.9: Relative error in the Az reconstruction as a function of frequency. Green and red curves are calculated taking into account decay and decoherence, but assuming a perfect qubit, i.e. the system is truncated to two levels. Green curve corresponds to resonant driving and red is to the off-resonant one. Blue and orange curves (resonant and off-resonant, respectively) are constructed from a full Transmon simulation with the Transmon truncated to the first ten levels.

characterization to higher frequencies. In a real experiment this method will eventually break down at higher frequencies too due to low signal contrast. Another major limiting factor of the experimental precision are the qubit’s decay and dephasing times which limits our precision at low frequencies. As the Rabi frequency starts to approach Γ1 and Γ2 it is not possible to resolve oscillations and our method fails. By comparison, from the simulation results between the multi-level transmon model and its two-level approximation, we conclude that the errors due the ﬁrst rotating wave approximation are not relevant for a transmon qubit compared to the errors due to the excitation of the higher levels. However, these errors may be important at higher Rabi frequencies for more anharmonic qubits. The errors due to the second rotating term are expected to be relevant at low frequencies but seem to be masked by the decay of the qubit. In conclusion, our numerical simulation shows that it is possible to apply our method in the frequency range of a standard AWG while keeping all systematic errors within 7% in amplitude and 0.2 radians in phase. The errors are dominated by population of the higher levels of the transmon and can be further reduced by starting oﬀ-resonant driving in expense of signal contrast. Extending the model to include the higher levels can also, in principle, mitigate the latter problem but will require more complex numerical procedures. We also note that applying our method to qubits with larger anharmonicity should extend the frequency range and result in a much more accurate reconstruction.

100 5.10. ACKNOWLEDGEMENTS

Transmon resonant sweep 0.6 Transmon off-resonant sweep qubit resonant sweep qubit off-resonant sweep 0.4 d

a 0.2 r

, z

0.0 n i

r o r

r 0.2 E

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5 Fitted Rabi frequency R, GHz

Figure 5.10: Error in the φz reconstruction as a function of frequency. Color-coding matches the plot above. One may note that even for a two-level case with higher detunings phase reconstructions gets less precise due to emergence of ﬁtting errors due to low signal contrast.

5.10 Acknowledgements

Note added – After this work was completed, we became aware of a complementary time-domain method [17] employing a qubit as an oscilloscope, which allows to sample control pulses of arbitrary shape making use of non-linear qubit frequency dependence on ﬂux.

101 BIBLIOGRAPHY

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103 BIBLIOGRAPHY

104 CHAPTER 6

MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

One of the factors limiting the fidelity of operations in superconducting quantum processors is the residual population of the excited state of qubits, which has routinely been measured to be many orders of magnitude higher than the one predicted from a Boltzmann distribution. Indeed, at the operation temperatures of dilution cryostats (≲ 20 mK) and with qubit frequencies around −5 48~20 5 GHz one might expect Pe ≈ e · < 1e-5, while the measured values might exceed one per cent [1–7]. The problem arising from such an increased effective qubit temperature Teff is twofold. First, ”The ability to initialize the state of the qubits to a simple fiducial state” is one of the DiVincenzo’s criteria for quantum computing, as many quantum algorithms require a known input state to effectively operate [8]. Second and no less important, high effective qubit temperature is an indication of existing extra channels of excitation events of a qubit. Examples of such channels in superconducting processors include stray radiation from higher stages of a dilution cryostat [1,9], not attenuated enough microwave noise in the control lines of qubits [10, 11], hot out-of-equilibrium quasiparticles [12–15] and even cosmic rays [14, 15]. A lot of efforts have been focussed in the field to address the first problem. The most natural way here is to use a qubit reset protocol to enhance the quality of a starting quantum state. It can also be used to significantly shorten the overall time required to execute a quantum algorithm, as waiting for the qubits to cool down to Sg⟩ state is inefficient with decay (T1) and decoherence (T2) times exceeding 100 µs [6]. A plethora of such protocols has been proposed in the last decade for a variety of computing platforms; The recent papers [16] and [6] offer two new and contain an overview of existing methods. Active and passive reset protocols, although increasing the algorithms’ fidelities and repetition rates, do little to combat the reasons behind the increased e-state population. A pathway here is to carefully and selectively engineer systems’ parameters while directly or indirectly monitoring the rate of spurious excitation events. For instance, in [13] the authors demonstrate their qubit (a 3D Transmon) to exhibit constant excitation and relaxation rates from quasiparticle tunneling events for temperatures less than 140 mK. They use the transmon in a regime QP QP where the rates of those events can be measured and find Γ↑ ≈ 1.12Γ↓ . In [17], a follow up paper to [13], an Eccosorb filter is added to the input-output port of the cavity, which decreases the quasiparticle tunneling rate from 4740 Hz to 166 Hz.

105 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

An instrumental tool to evaluate the reset methods and develop new ones, quantify the quality of the state initialization and, more importantly, to identify and eliminate the sources of spurious excitation is the ability to resolve the changes in the excited state population of a qubit within fractions of a per cent and in a short period of time. Two routinely applied methods make use of high signal-to-noise ratio (SNR) or the third level of a qubit (Sf⟩-state). A high SNR enables one to perform a single-shot readout (i.e. measuring the state of the qubit with high fidelity without averaging) with high contrast and simply count the resulting occurrence of e-state in the prepared ground state. Having a third level enables observing Rabi-type oscillations between the Se⟩ and Sf⟩ states of the system and comparing the signal with the ’reference’ one, with the ratio giving the fraction of Se⟩-state [4, 18]. For some systems, however, due to coupling or Q-factor restrictions it is not possible to reach the required single-shot SNR even in the presence of quantum-limited amplification; With no QL-amps, such as JPA or TWPA, usual SNRs are significantly lower than it is required to do single-shot. An alternative method cannot be applied if the higher levels are not accessible due to large discrepancy of transition frequencies or selection rules [19]. Finally, both methods have limiting error sources whose magnitude does not depend on the effective temperature being measured and therefore poses an absolute precision floor. In this chapter we introduce a novel method allowing to measure the excited state (Se⟩-state) population (or effective temperature Teff ) of a qubit using correlations between two sequential measurements. Utilizing the quantum non-demolition (QND), i.e. projective, nature of the measurement, we lift the requirements for high-fidelity single-shot readout or for manipulations involving higher levels of the system employed as a qubit to measure its effective temperature. We analyse the error budget of the method and show that it does not have the absolute precision floor; Its accuracy limit is not defined by SNR and is only sensitive to qubit decoherence and gate errors in the second order. Because of that we achieve the highest reported precision of the excited state population measurement with accuracy of 0.01% and study its dependence on the qubit transition frequency. Although our experimental demonstration is carried out on the platform of circuit quantum electrodynamics (QED) and transmon qubits, the method is generic and is applicable to any system where the QND measurement can be realised. The results of this chapter are now published in Phys. Rev. Lett. 124, 240501 (2020) [20].

6.1 Experimental platform

Our experimental system is similar to the one used in [21] and consists of a tunable-frequency transmon coupled to a 3D microwave cavity. The cavity is employed to both carry the microwave pulses to manipulate the qubit and to readout its state. The transmon has a weakly anharmonic multi-level structure, and its two lowest energy eigenstates are used as the logical states Sg⟩ and Se⟩ of a qubit. We are also using the next energy eigenstate Sf⟩ to realize the qutrit protocol mentioned above [4, 18] for comparison. The system is tuned to the dispersive regime, where the qubit Sg⟩-Se⟩ transition frequency is far from the cavity transition frequency, so the standard

106 6.2. THEORETICAL IDEA dispersive readout method can be employed [22]. For low readout powers the dispersive readout has highly quantum non-demolition nature with negligible contribution to qubit excitation due to readout. To achieve a high SNR and the ability to readout a transmon state in a single-shot regime we use a Josephson parametric amplifier (JPA) similar to one described in Ref. [23]. It provides us with quantum-limited amplification and results in SNR of ∼ 6, which is sufficient to differentiate between three transmon’s states with high fidelity, allowing us to ensure the low measurement backaction and realize another (direct) Teff measurement technique mentioned earlier. We do not, however, modify the dispersive readout technique with shifting the probe tone frequency as discussed in [21] (and the previous section) as for the correlator method measurements the signal visibility between Sg⟩ and Se⟩ is of paramount importance.

6.2 Theoretical idea

We ﬁrst present the idea of our method using a notion of an abstract ideal quantum two-level system with an instant and noiseless quantum non-demolition measurement. We can deﬁne the measurement apparatus to yield a real value Vg for the qubit in the ground state and Ve for the qubit in the excited state. By repeating the same experiment many times the average value of the measurement response is expressed as

(0) ⟨V ⟩ = g = PgVg + PeVe ≡ V˜g, (6.1)

where Pe is spurious Se⟩-state population, Pg = 1 − Pe is the ground state population and g(0) is zero-th order correlation function. Knowledge of ⟨V ⟩ can be in principle suﬃcient to determine the excited state population Pe if the responses Vg~e are known. Unfortunately, these responses are generally not known a priory and to determine Pe one needs to make additional measurements such as some measurements involving the second excited level [4, 18]. Instead of using the higher excited levels we propose to measure the ﬁrst order correlation function g(1)(τ) = ⟨V (0)V (τ)⟩. Assuming that our measurement is quantum non-demolition (QND) the second subsequent measurement will return a fully correlated result

(1) 2 2 g (0) = PgVg + PeVe . (6.2)

This value can be compared to

2 g(1)(∞) = g(0) ≤ g(1)(0), (6.3) where we assumed that the measurements will be fully uncorrelated if separated by long times. It is also straightforward to see that the equality g(1)(0) = g(1)(∞) is realized only if the qubit is its ground state Pg = 1 (or Pe = 1).

107 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

1.4 67 mK (T1 = 9.37 µs) 22 mK (T = 11.84 µs) 1.2 1

1.0

0.8 ) (%) τ

( 0.6 (1) ¯ g 0.4

0.2

0.0 0 10 20 30 40 50 τ (µs)

Figure 6.1: Decay of the normalized correlator between two sequential measurements separated by τ. The solid lines represent exponential T1-decay. Amplitude of the correlator at zero (or lowest attainable) delay allows one to reconstruct the Se⟩-state population and hence the eﬀective temperature of the qubit.

Figure 6.2: The experimental protocol. “Run I” represents measurement of the correlation function (1) (0) g (τ) and g . “Run II” is an additional calibration measurement required for correct scaling of Pe. (1) The variable delay was used to measure the decay of g (τ). To determine Pe only one measurement with τ = 0 is necessary.

Measurement of a typical decay of the correlation function is shown in Fig. 6.1; It follows an exponential curve with the relaxation time T1 of the qubit. Observation of this decay is the manifestation of the spurious Se⟩-state qubit population. However, to determine Pe quantitatively we need to add a calibration measurement. For example, we can apply a π pulse to swap the ground and excited state populations before taking a measurement (see Fig. 6.2) returning (0) gπ = PeVg + PgVe ≡ V˜e. (6.4)

Using simple calculations and an assumption of Pe being small (see below) one can obtain

2 g(1)(0) − (g(0)) Pe ≃ » 2 . (6.5) (0) (0) (1) g + gπ − 2 g (0)

108 6.2. THEORETICAL IDEA

6.2.1 Approximate expression derivation

Equation (6.5) can be derived from the following system of equations:

⎧ 2 2 (1) ⎪Pg · Vg + Pe · Ve = g (0) ⎪ ⎪ 2 2 + + 2 2 = (1)(∞) ⎪Pg · Vg 2PgPeVgVe Pe · Ve g ⎨ 2 ⎪1 2 2 1 (0) (0) (1) ⎪ (Vg + 2VgVe + Ve ) = g + gπ ≡ g (∞) ⎪4 4 π~2 ⎪ ⎩⎪Pg + Pe = 1

(1) First-order correlator gπ~2(∞) in the third equation above can be also obtained by applying a π~2 pulse to equate the populations of the ground and excited states before taking the measurements, thus the notation.

Assuming that Pe is small we write the system to the lowest order in Pe as:

⎧ 2 2 2 (1) ⎪V + Pe(V − V ) = g (0) ⎪ g e g ⎪ 2 2 (1) ⎨Vg + 2Pe(VgVe − Vg ) = g (∞) (6.6) ⎪ ⎪( + )2 = (1) (∞) ⎩⎪ Vg Ve 4gπ~2

The diﬀerence of two correlators reads

g(1)(∞) − g(1)(0) =

Pe [2Vg(Ve − Vg) − (Ve − Vg)(Ve + Vg)] =

= Pe(Ve − Vg)(Vg − Ve) ≃ (6.7) ½ » 2 (1) (1) ∼ Pe 2 gπ~2(∞) − 2 g (0) ,

» (1) where we utilized the smallness of Pe to write Vg ∼ g (0) and used the third equation from (6.6). It follows that g(1)(0) − g(1)(∞) ≃ Pe ¼ » 2 . (6.8) (1) (1) 4 gπ~2(∞) − g (0)

Note, that only measuring g(1)(0) requires actual correlation of two sequential readouts. (1) (1) Both g (∞) and gπ~2(∞) can be obtained by squaring the mean values of the ground state and 50/50 mixture of Sg⟩ and Se⟩ states, respectively.

6.2.2 Normalizing the responses

The amount of averages N required to reach a certain ﬁnal precision for a ﬁrst order correlation function scales as the single measurement SNR (denote r) to a power of minus four N ∝ r−4. This scaling is unfavorable compared to N ∝ r−2 for a zeroth order correlation function [24] and requires additional measures to optimize the SNR.

109 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

0.2

V˜g 0.1 0.2 0.1 0.0 0.0 ˜ 0.1 Ve −

Q (arb. units) 0.1 0.2 − − 0.2 0.0 0.2 −

0.2 − 0.1 0.0 0.1 0.2 0.3 0.4 − I (arb. units)

Figure 6.3: Single-shot measurement data with no pulse (π pulse) shown as blue (red) pixels. The white (black) circle shows the averaged response V˜g (V˜e). Dashed is the line joining Vg and Ve; We assume deviations perpendicular to this line are only caused by noise, and project the complex IQ measurement onto it.

We first used a quantum limited JPA to amplify the signal to a high SNR of ≈ 6 in order to classify the measurement outcomes in a single shot. After performing single shot measurements and state classification, the data is converted to the binary form (Sg⟩ → 0; Se⟩ → 1). The procedure allows for measurement of decay of the correlation function g(1)(τ) and extraction of the excited state population without the noise of the amplification chain. As expected, the result coincided perfectly with the direct counts of Sg⟩ and Se⟩ states. In order to make the method work without the state classification, we use the integrated heterodyne voltage V to extract a real-valued response containing a maximum amount of information. First, we do a standard set of calibration measurements to determine V˜g and V˜e – the averaged measurement responses for prepared Sg⟩ and Se⟩ states – which are complex values of integrated heterodyne voltage in the circuit QED case. Each of the subsequent measurements we project onto the line containing the two calibration responses, so that V˜g → 0 and V˜e → 1 (see Fig. 6.3). This is done simply by the following transformation:

V¯ = Re (V − V˜g)~(V˜e − V˜g) . (6.9)

(0) The zero-th order correlation functions of V¯ are of a particularly simple form:g ¯ ≡ PgV¯g + (0) PeV¯e = 0 andg ¯π ≡ PeV¯g + PgV¯e = 1. Therefore the Eq. (6.8) simpliﬁes to

g¯(1)(0) Pe ≃ » » 2 , (6.10) 4 1~4 − g¯(1)(0)

(1) ¯ ¯ (1) (1) whereg ¯ (0) ≡ ⟨V (0)V (τ)⟩Sτ=0, and we usedg ¯ (∞) = 0 andg ¯π~2(∞) = 1~4. Moreover, after the transformationg ¯(1)(0) is close to zero since the qubit is mostly in the ground state. Any

110 6.3. RESULTS error in the estimation ofg ¯(1)(0) will be ampliﬁed due to the square root operation of a small » number in the expression for excited state population (Eq. (6.10)). Removing g¯(1)(0) in the denominator will not lead to inaccuracy for a small excited state population but will greatly (1) reduce noise ing ¯ (0). These manipulations lead to the following expression for Pe:

(1) Pe ≃ g¯ (0). (6.11)

Alternatively, one can re-derive the expression for Pe using the normalised responses and obtain an exact solution for the system of equations with three unknowns:

⎧ ¯ 2 ¯ 2 (1) ⎪Pg · Vg + Pe · Ve = g¯ (0) ⎪ ⎪ 2 ¯ 2 + ¯ ¯ + 2 ¯ 2 = ⎪Pg · Vg 2PgPeVgVe Pe · Ve 0 ⎨ ⎪1 ¯ 2 ¯ ¯ ¯ 2 ⎪ (Vg + 2VgVe + Ve ) = 0.5 ⎪4 ⎪ ⎩⎪Pg + Pe = 1

Note that after the normalization, V˜g → 0 and V˜e → 1 but the ’true’ responses Vg → V¯g and

Ve → V¯e are still unknown. Solving the system leads to a precise expression:

1 1 (1) Pe = − » ≃ g¯ (0), (6.12) 2 2 1 + 4¯g(1)(0) where the approximation holds when Pe ≪ 1, thus ensuring the consistency of the precise expression with the approximate one derived above.

6.2.3 Added measurements noise

In reality, measurement of the correlation function returnsg ¯(1)(τ) = ⟨V¯ (0)V¯ (τ)⟩ + ⟨η(0)η(τ)⟩, where η includes contributions of all noise sources such as noise of the amplification chain and the quantum noise. For a typical experimental setup the measurement noise is “fast” and ⟨η(0)η(τ)⟩ = 0 for all relevant time scales. The noise contribution can be suppressed by acquiring sufficient statistics for all τ > 0. The noise contribution at τ = 0 can be, in principle, subtracted by performing additional calibration measurement of ⟨η2⟩. In our experiments, we simply approximatedg ¯(1)(0) by a correlator of the results of two sequential measurement in time (see Fig. 6.2). Systematic study of the standard deviation ofg ¯(1)(0) shows the expected scaling with a number of averages N up to N = 216 confirming the absence of any measurable “slow” noise contribution in our measurement setup (see below).

6.3 Results

We have performed a study of residual excited state population of a Transmon qubit vs the temperature of the mixing chamber (MC) plate of a dilution refrigerator shown in Fig. 6.4. For each temperature point after stabilizing the MC sensor temperature we have waited ample

111 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS time (> 1 hour) for the qubit and its environment to thermalize and performed measurement of the qubit Se⟩-state population using four diﬀerent methods for each MC temperature point. First, we used our method in the presence of a quantum-limited ampliﬁer (JPA), which gives us a fairly high SNR of ∼ 6 and allows determining the residual Se⟩-state population with the precision of .01% in 15 minutes, which is the highest precision reported [7, 18]. Interestingly, the standard deviation of our method was smaller than the direct counting of excitations using the same data.

6 (a) 5 0.4

4 0.3 3

(%) 20 30 40

e 2 P

1 0.33% 0

1 − 20 30 40 50 60 70 80 90 100 (b) 2 1 0 2 JPA on 1 0 (%) JPA oﬀ

e 2

P 1 0 counts 2 1

0 qutrit 1 − 20 40 60 80 100 MC Temperature (mK)

Figure 6.4: (a) Measured Se⟩-state population as a function of the mixing chamber sensor temperature. Red points are correlator measurements with a JPA. Data for each point corresponds to 220 repetitions. The blue points are measured with JPA turned off. The black solid line corresponds to the M-B distribution offset by 0.33% as indicated by the dashed green line. The error-bars cover two standard deviations in measurement (95% confidence). (b) Deviation of the data from the M-B distribution for different methods (see text for more details).

In the second measurement we used our method without the JPA. It resulted in a SNR of 0.9 which is not suﬃcient for a single-shot measurement. The results were in agreement with the precise measurements, thus demonstrating the ability of our method to work in the conditions of low SNR (Fig. 6.4a). We have also used conventional methods to determine the Se⟩-state population using the second excited state of the Transmon and the direct count of single shots

112 6.3. RESULTS

1.4

1.2

1.0

0.8 (%) e

P 0.6

0.4

0.2

0.0 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 Qubit Frequency (GHz)

Figure 6.5: Excited state population vs qubit frequency representing a “noise spectrum” as seen by the qubit. The green arrow indicates the qubit frequency used for the rest of the experiments. making use of JPA [23]. All methods’ results are in agreement within the error bars but show different statistical and systematic errors (see Fig. 6.4b and below for more comments). The residual Se⟩-state population of our qubit as function of the temperature of MC plate coincides within the error bars (< 0.01% uncertainty) with the M-B curve shifted by a “zero- temperature excitation” offset (the curve is indicated on the plot with a solid black line). Note that both the offset value and the M-B distribution have no free parameters: the offset is given by the measurement at the lowest attainable temperature and the qubit transition energy was obtained independently using spectroscopy and Ramsey-type measurement. Our results are somewhat different from the conclusion of Ref. [18] where spurious excitation followed the M-B distribution without an offset, but saturated at the temperature of 35 mK. The presence of this offset may be explained by a model of a qubit being coupled to two separate thermal baths. One of the baths is strongly coupled to the qubit and thermalised with the MC plate of a dilution refrigerator, while the second bath is weakly coupled but has a much higher temperature independent of the MC temperature. We determined the rate of excitation and relaxation events from this second, non-equilibrium source, to not exceed 670 Hz, corresponding to a time constant of 1.5 ms which is consistent with ’hot’ out-of-equilibrium quasiparticles as a possible origin for the qubit excitation [13, 25]. To acquire more information on the origin of the qubit excitation we used our method to perform “temperature spectroscopy” by measuring the Se⟩-state population as a function of the qubit frequency. Figure 6.5 shows that the Se⟩-state population peaks around 6 GHz and can change abruptly with even small changes in the qubit frequency. This behaviour is inconsis- tent with the excitation by quasiparticles whose matrix element is a smooth function of qubit frequency [26]. Instead, this behaviour is characteristic to coupling to two-level systems (TLS) which are believed to be the dominant source of the qubit relaxation and exhibit a strong non- monotonic dependence of relaxation times of superconducting qubits on their frequencies [27].

113 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

1.4

1.2

1.0

0.8 (%) e

P 0.6

0.4

0.2

50 Qubit Frequency (GHz)

) 30 µs (

1 20 T

0 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 Frequency (GHz)

Figure 6.6: a) Excited state population and b) Relaxation time vs qubit frequency. The green arrow shows the frequency used in the rest of the experiments. The black dashed arrows indicate the simultaneous jumps in population and relaxation time, indicating that they are related. The purple line is the Purcell limit of relaxation time.

6.3.1 Relaxation time vs Frequency

Figure 6.6 shows the excited state population and relaxation time as a function of qubit frequency. There is a visible increase in population at frequencies between 5.75 to 6 GHz. More- over, the population changes drastically on a scale of 100 kHz (comparable to a linewidth of the qubit). This is characteristic of T1 frequency dependence where the relaxation is dominant by TLS and cannot be explained by quasiparticle tunneling or spurious electromagnetic modes [27]. Some correlation between jumps in population and relaxation time change are indeed visible (shown by the black dotted arrows). Unfortunately, T1 is severely limited by the Pur- cell limit at higher frequencies which may mask the TLS contribution to the relaxation time. If TLS are responsible for this anomalous population increase it would suggest that they are even at higher temperatures than the qubit. Possible sources of TLS heating may include hot quasiparticles [13] which are generated in the capacitor pads of the transmon and dissipate their energy before reaching the Josephson junctions. However, further research is necessary to identify the exact source of the anomaly.

114 6.4. PRECISION AND ERRORS

6.4 Precision and errors

This section is devoted to the analysis of both systematic and statistical errors of the correlator method and the two conventional ones. Since we are measuring a quantity which magnitude is unknown a priori, it is not possible to compare the result of each of the methods to some ’true’ reading. Instead we discuss the possible error sources, methods’ sensitivity to them and what sources lead to a bias in the results and an eventual precision floor on the accuracy of the methods. We are going to distinguish two sorts of errors with respect to precision: – Relative errors, i.e. errors causing under- or overestimation of the result by some fraction. These errors may limit the relative precision of a method, but they do not pose an absolute precision floor. – Absolute errors, i.e. errors which do not depend on the Se⟩-state population of a qubit, but rather on other setup parameters, such as the system’s SNR, measurement time or setup stability. While of course the magnitude of the errors under consideration is the primary factor, we generally note that the absolute errors are more detrimental as they limit both relative and absolute precision and eventually lead to a limit on the lowest measurable Teff . Relative errors, on the other hand, in principle allow distinguishing arbitrary low Se⟩-state populations.

6.4.1 Direct counting

Direct counting of the occurrences of Se⟩ state in the prepared ground state of a qubit is the easiest and the most natural way to obtain the Se⟩-state population, and effective temperature, of the qubit. It does not require using any control pulses and was very instructive for a reliable verification of our method. With high signal-to-noise ratios of 10+ it offers the direct and the most reliable way to do so while also allowing one to immediately see and quantify all error sources. Unfortunately, direct counting is only possible for a readout with sufficiently large SNR. With lower SNR the absolute error due to state misinterpretation rises exponentially thus limiting the practicality of this method for temperature measurement, especially for very small spurious populations. Its statistical error scales as the one for a single measurement and therefore decays quickly with the amount of repetitions. A figure here would be the standard deviation of the amount of Se⟩ counts in the prepared ground state, and therefore it naturally does not have a noise floor.

There are three main causes of systematic errors, however: normal T1-decay (or excitation) during the readout, measurement backaction and misinterpretation of the outcomes falling outside of the state line boundary.

The T1-decay possesses two important properties allowing us to integrate it into the model and disregard entirely. First, it is a relative error and can be estimated precisely both in case of decay and excitation. And second, in case of equilibrium Se⟩-state population the rate of decays equals the rate of excitations, and therefore this error does not play any role in the total error

115 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS budget. Misinterpretation is another straightforward error to consider qualitatively. It is obvious that this error is absolute, as it does not depend in the first order on the Se⟩-state population, but depends only on the SNR. As an example, if the SNR is 6, the decision boundary between g-state and Se⟩-state would be at the distance of three sigmas from the ”true” values of both Sg⟩ and Se⟩, which would lead to an amount on the order of ”three sigma error” – that is, 0.3% in this case – to be misinterpreted. Of course, it is possible to employ various techniques to mitigate those errors somehow, but the key property – the errors being absolute – stays. Non-Gaussian noise distribution due to multiplied noise from the amplification chain and room-temperature elements presents an extra complication in realization of the mentioned mitigation techniques. Extra assumptions on the origins and distribution of the readout noise should be used, which can prove to be a challenging task. To sum up, the statistical error following this way is sufficiently small after only a few repetitions, and generally one just requires an ample amount of Se⟩ counts to infer the population. From this we can say that statistical error scaling in this way is the best out of the three presented methods. However, the systematic errors due to misinterpretation of the outcomes pose an eventual absolute precision floor on the accuracy of the method.

6.4.2 Qutrit

In addition to the requirement of going out of the qubit subspace and exploiting the higher (Sf⟩) level of a physical system, the qutrit method suffers from a few extra error channels which are largely absent in both the correlator and single-shot method cases. The small errors ε of calibration pulses, such as π pulse in the correlator method or the π pulse between Sg⟩ and Se⟩ levels in the qutrit method, contribute to the total error budget only in the second order and create a relative error. It is straightforward to see, as those errors go to the denominator of the corresponding expressions to determine the Se⟩-state population, thus multiplying the scale of the measurement by a factor 1 + ε close to unity. An Se⟩-Sf⟩ π pulse in the qutrit method, however, creates a small first-order error by directly exciting the Se⟩ state of the qutrit due to the coupling of the drive to the Sg⟩-Se⟩ transition. Another reason is the envelope of the Se⟩-Sf⟩ pulse in the frequency domain, which has finite bandwidth commensurate with the anharmonicity of the qubit (usually on the order of tens per cent). This unwanted excitation of Se⟩ state during the measurement of its population is an example of a leakage error [28] present in the qutrit protocol. Therefore calibrating the Se⟩-Sf⟩ pulse so that in the Fourier space no Sg⟩-Se⟩ frequency component is present becomes imperative for the qutrit method in order to measure low populations. While certain optimal control techniques, such as DRAG pulses [29] for Se⟩-Sf⟩ transition, could be employed to mitigate this problem, it is not possible to entirely isolate the frequencies and therefore it poses an extra limitation on the precision of the method. Note that this error is absolute and systematic, and the decrease of statistical uncertainty can not guarantee its negligible magnitude or give information about its value.

116 6.4. PRECISION AND ERRORS

Fig. 6.5 shows an increase in the eﬀective temperature of the qubit slightly below 6 GHz, which we attribute to coupling to two-level systems. This conclusion is supported by previously observed anticrossings of the qubit with TLSs at these frequencies, yet more investigation is required to determine its source with certainty. The coupling of the Se⟩-Sf⟩ transition to two- level ﬂuctuators leads to instability in its frequency and hybridisation of the transition, thus enhancing its decoherence and preventing the statistical error from decreasing. It might be a primary reason of the largest absolute discrepancies in measured e-state population showed by the qutrit method. It is interesting to note that the qutrit method demonstrated the worst accuracy which may be attributed to extra decoherence due to Sf⟩-level and to the direct excitation of Se⟩-state when applying Se⟩-Sf⟩ drive. While certain optimal control techniques, such as DRAG pulses [29] for Se⟩-Sf⟩ transition, could be employed to mitigate this problem, impossibility to entirely isolate spurious Se⟩-state excitation by the method itself poses an extra limitation on its absolute precision.

6.4.3 Correlator method

The largest systematic error source of our method comes from the ﬁnite time of the measurement, which leads to a partial decay of the correlations following the standard T1 decay curve

(see Fig. 6.1). While this error can be considerable, a separate measurement of T1 can be used to correct for this error. Most importantly, this error is relative, as it only decreases the −T ~T measured Pe by a factor of e meas 1 , where Tmeas is the measurement time. Therefore, this error does not set a lower limit on the measurable spurious population unlike the error of the ﬁnite SNR for the direct counting.

A similar effect is due to π pulse errors. As this error only affects V˜e which is measured independently from g(1), it only contributes as a relative error and does not affect statistical distribution for Pe. Moreover, if an infidelity of the π pulse is small this error contributes to Pe only in the second order. Similarly to the direct counting method, measurement of g(1) does not involve any control pulses and is generally performed when the qubit is in equilibrium with environment. Therefore, the only possible systematic absolute error of our method arises from the excitation of the qubit due to dispersive readout which can be virtually arbitrarily suppressed by larger qubit detunings and/or lower readout powers. The only statistical (not systematic) error of our method is due to measurement noise which, in turn, can be reduced by increase in averaging time. Fig. 6.7 shows a standard deviation of measured Se⟩-state population as a function of number of measurements and different readout powers. The error scales as N −1~2, where N is the number of iterations, over the complete range deviating from this expected dependence only for the largest power of -30 dBm, most probably, due to loss of quantum non-demolition behaviour of the readout.

117 CHAPTER 6. MEASURING EFFECTIVE TEMPERATURES OF QUBITS USING CORRELATIONS

10 2

8 1 (%) e 6 P 0

7.5 10.0 12.5 15.0

) 4 e /P e 2 δP ( 2

log 0 -48 dBm -45 dBm 2 − -42 dBm -39 dBm -36 dBm 4 − -33 dBm -30 dBm 6 − 4 6 8 10 12 14 16 log2(iterations)

Figure 6.7: Relative precision of Pe measurements and their linear ﬁts. The precision scales as expected for uncorrelated noise (indicated by the black dashed line). Inset: Population (solid line) and standard deviation (ﬁll) for a measurement power of −30 dBm.

6.4.4 Slow noise contributions

If a sequence of measurement segments requires acquiring enough statistics, we perform the sequence in the interleaved regime. It means we repeat the entire sequence N times and average afterwards, rather than repeating the ﬁrst segment for N times before moving to the next one. This ensures that various slow compared to the length of a single repetition drifts in output signal amplitude and phase are being averaged out as they appear in all segments equally. This is also true for the normalization procedure for the correlator measurement (involving the transformation described in Eqn. 6.9). However, for a correlation measurement the slow noise contributions can no longer be hidden by interleaving, as the slow noise is squared and does not average to zero: ⟨noise⟩ → 0, but ⟨noise2⟩ ↛ 0. This could potentially lead to a “noise

floor” that limits the precision of Pe measurement. We plot the precision of the measured population as a function of the number of measurements in Fig. 6.7. Evidently it does not hit a noise floor for the number of measurements used, thus demonstrating the stability of the setup used. A possible approach to decrease the effect of this slow noise is to split the measurements into many buckets and calculate the population separately for each bucket. The final result would be

118 6.5. DISCUSSIONS the mean of the populations extracted from each bucket. There is a natural trade-off between the slow and fast parts of the noise spectrum: if the number of measurements in a bucket is too low, the calculations leading to the e-state population may break down. For example, the » g(1)(0) term in the denominator of (6.10) rendered the expression for the population invalid when it was negative. Its value is close to zero for low populations for the normalized data; Removing it in (6.11) or using precise equation (6.12) stabilizes the expression. High number of measurements in a bucket is potentially more affected by slow noise. Again, we were unable to see statistically significant differences in the results obtained by direct calculations performed on the whole data and by using the bucketing approach. Using thin-film attenuators [10], cavity attenuators [30] or Eccosorb filters [17] in the microwave lines that lead to the system have been shown to reduce the excited state population of 3D Transmon qubits. We do not use any of these in our experimental setup. Additionally, engineering the superconducting gap [31–33] by varying the thickness of the deposited super- conductor has been shown to trap quasiparticles and reduce their tunneling rate which reduces the qubit population. Our qubit was not fabricated with such quasiparticle traps. However, our qubit at the base temperature of 20 mK has a lower population than a standard circuit QED setup. One possible explanation for this is that we use three isolators on the output of the cavity, with a total isolation of 54 dB.

6.5 Discussions

Discussions.— In summary, we have proposed and experimentally realized a method of measuring the eﬀective temperature of qubits using correlations between consecutive measurements. Our method does not require usage of higher excited levels, is less susceptible to errors in control pulses and allows for virtually unlimited suppression of absolute errors even without high SNR required for the high-ﬁdelity single-shot measurement. Our method can be used on any platform; We experimentally show it to have the highest reported precision for superconducting circuits. The accuracy of our method enables “temperature spectroscopy” giving spurious population of Se⟩-state of the qubit as function of qubit transition frequency which can shed light on the sources of decoherence.

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