ADVANCED DYNAMICAL DECOUPLING STRATEGIES FOR SIMULATINGHAMILTONIANINTERACTIONS Vom Fachbereich Physik der Technischen Universität Darmstadt zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von M. Sc. Holger Frydrych aus Lauterbach Referent: Prof. Dr. Gernot Alber Korreferent: Prof. Dr. Reinhold Walser Tag der Einreichung: 12.10.2015 Tag der Prüfung: 14.12.2015 Darmstadt 2016 D17 ABSTRACT In the field of quantum information, a major challenge is to protect and shield quantum systems from environmental influences, which cause dissipation and decoherence. One approach to address this problem is called dynamical decoupling, where we act on a quan- tum system with a series of fast and strong local control operations. If done correctly, the system’s state gets rotated in discrete steps in its state space in such a way that the effects of the environment cancel up to a certain order. In the first part of this thesis, we will introduce the basics of dynamical decoupling and present a new approach for con- structing dynamical decoupling schemes based on sequences of Weyl operators on networks of qudits. This approach goes beyond merely protecting a quantum system, as it is also capable of altering existing Hamiltonian interactions in ways suitable for quantum simulation purposes. We will also investigate how imperfect controls influence the effectiveness of dynamical decoupling and focus particularly on stochastic noise. In the second part of the thesis, we study concrete scenarios for the application of dynamical decoupling. First, we consider the task of quantum state transfer on a linear chain of qubits and use decou- pling to protect the transfer from detrimental influences caused by a bend in the chain. We also look at how to design the specific interac- tion strengths required on the chain to make the state transfer work and extend this result to chains of qudits. Another chapter deals with decoupling the atomic centre-of-mass motion of a trapped atom or ion in a cavity interacting with a radiation field. Finally, we use de- coupling to execute sequences of one- and two-qubit quantum gates on a chain of coupled qubits. Particular care is taken to ensure high fidelity operations in the presence of imperfect decoupling pulses. iii ZUSAMMENFASSUNG Eine der größten Herausforderungen in der Quanteninformations- theorie ist der Schutz von Quantensystemen vor Umgebungseinflüs- sen, die Dekohärenz und Dissipation verursachen. Eine mögliche Lö- sung ist Dynamische Entkopplung. Hierbei wird das zu schützende Quantensystem einer Serie von schnellen, lokalen Kontrolloperatio- nen ausgesetzt, die bei korrekter Anwendung den Zustand des Sys- tems im Zustandsraum so rotieren, dass sich die Umgebungseinflüsse im Mittel bis zu einer bestimmten Ordnung aufheben. Im ersten Teil dieser Dissertation werden die Grundlagen dynamischer Entkopp- lung erklärt und anschließend ein neues Verfahren für die Konstruk- tion von Entkopplungssequenzen auf Qudit-Netzwerken vorgestellt, deren Kontrolloperationen aus Tensorprodukten von Weyl-Operatoren bestehen. Dieses Verfahren ermöglicht auch Entkopplungssequenzen, die nicht dem reinen Schutz eines Quantensystems dienen, sondern bestehende Wechselwirkungen für den Zweck der Quantensimulati- on manipulieren können. Abschließend wird auf den Einfluss von imperfekten Kontrollen auf die Effizienz von Entkopplungsverfahren eingegangen und dabei insbesondere der Effekt von stochastischem Rauschen analysiert. Der zweite Teil der Dissertation beschäftigt sich mit konkreten An- wendungsszenarien für dynamische Entkopplung. Zunächst wird die Zustandsübertragung auf einer linearen Qubit-Kette betrachtet, wo- bei Entkopplung verwendet wird, um den Transfer vor den Einflüs- sen eines Knicks in der Kette zu schützen. Anschließend wird gezeigt, wie die für die Zustandsübertragung notwendigen Wechselwirkungs- stärken mittels Entkopplung konstruiert werden können. Dieses Re- sultat wird dann auch auf Ketten von Qudits verallgemeinert. An- schließend wird die Wechselwirkung eines gefangenen Atoms oder Ions in einer optischen Kavität mit einem Strahlungsfeld betrachtet und aufgezeigt, wie die Bewegung des Atoms in der Falle entkop- pelt werden kann. Abschließend wird dynamische Entkopplung da- zu verwendet, Sequenzen von Quantengattern auf einer stark wech- selwirkenden Qubit-Kette zu implementieren. Hierbei wird insbeson- dere darauf eingegangen, wie trotz imperfekter Entkopplungspulse die Gatter mit hoher Qualität ausgeführt werden können. iv ACKNOWLEDGMENTS I would like to thank Prof. Dr. Gernot Alber for inviting me into and giving me the opportunity to work with his group and for his continued support during the writing of this thesis. I would also like to thank the members of the group for many fruitful discussions and in particular J. Zsolt Bernád, with whom I have had several success- ful collaborations on dynamical decoupling. Without him, the chap- ters about the influence of stochastic noise and about decoupling the centre-of-mass motion would not exist. He also graciously agreed to proof-read my thesis. Furthermore, I would like to extend my gratitude to Prof. Igor Jex and his group in Prague. I was granted numerous opportunities to visit them and benefited greatly from the experience. In particular, I would like to thank Pavel Bažant who helped me understand the basics of dynamical decoupling and who helped me develop my idea for the construction method presented in this thesis. I would also like to thank Antonín Hoskovec, with whom I studied and developed several of the ideas about state transfer in Ch. 6. Prof. Dr. Reinhold Walser graciously agreed to be the second as- sessor for this thesis, for which I am very thankful. Finally, I am grateful to the Crossing project, a collaborative SFB project focussed on cryptography, which I was given the opportunity to join in October 2014 and which paid for half of my salary and several conference visits. v CONTENTS 1 introduction 1 1.1 Outline 4 1.1.1 Theoretical foundations of dynamical decoupling 4 1.1.2 Practical applications of dynamical decoupling 5 1.2 Fundamentals 6 1.2.1 Quantum systems, states and observables 6 1.2.2 Measurement of physical variables 7 1.2.3 Evolution of a quantum system 8 1.2.4 Heisenberg and interaction picture 9 1.2.5 Composite quantum systems 10 1.2.6 Qubits and the Pauli operators 11 1.2.7 The quantum harmonic oscillator 12 1.2.8 Fidelity measures 12 i theoretical foundations of dynamical decoupling 15 2 overviewofdynamicaldecouplingmethods 17 2.1 Principles of dynamical decoupling 18 2.1.1 Average Hamiltonian and the Magnus expan- sion 19 2.1.2 Basic control strategies 21 2.2 Decoupling with instantaneous pulses 21 2.3 Higher-order decoupling strategies 23 3 designingpaulipulsedecouplingsequences 25 3.1 Networks of qubits and decoupling with Pauli pulses 26 3.2 Constructing Pauli decoupling schemes 27 3.2.1 Constructing a linear set of equations from the decoupling condition 27 3.2.2 Existence of solutions 29 3.2.3 Practical considerations 32 3.3 Example applications 33 3.3.1 Protecting a two-qubit interaction from environ- mentally inducedp decoherence 33 3.3.2 Protecting a SWAP gate implementation 34 3.3.3 Protecting arbitrary two-qubit interactions on an N-qubit network from environmental influences 35 3.3.4 Removing diagonal couplings in a closed 4-qubit chain 37 3.3.5 Modifying individual interaction strengths 38 3.4 Numerical simulations 39 4 extension to higher-dimensional systems 43 4.1 Generalizing the Pauli operators 43 vii viii contents 4.1.1 The generalized Gell-Mann operators 44 4.1.2 The generalized spin operators 45 4.2 Deriving the linear set of equations for qudits 45 4.2.1 Basic spin operator properties 46 4.2.2 Deriving the linear set of equations 47 4.2.3 The properties of the system matrix A(N) 48 4.2.4 The particular solution and construction of a decoupling scheme 49 5 imperfections in decoupling controls 51 5.1 Self-decoupling control sequences 52 5.1.1 The error of bounded controls 52 5.1.2 A single-qubit self-stabilizing decoupling sequence 53 5.1.3 Eulerian path decoupling 55 5.1.4 Higher-order self-decoupling sequences and nu- merical results 56 5.2 Influence of stochastic noise on dynamical decoupling 58 5.2.1 Decoupling pulses with stochastic noise 59 5.2.2 Time evolution with stochastic noise 60 5.2.3 The limit of continuous control 61 5.3 Examples with stochastic noise 62 5.3.1 Two coupled qubits 63 5.3.2 A spin interacting with a bath of nuclear spins 67 5.3.3 Two coupled harmonic oscillators 70 ii practical applications of dynamical decoupling 73 6 statetransferonqubitandquditnetworks 75 6.1 Perfect state transfer on qubit chains 76 6.2 Correcting errors in bent qubit chains 78 6.2.1 Complete selective decoupling scheme 80 6.2.2 Partial selective decoupling scheme 83 6.2.3 Disorder-resistant decoupling scheme 86 6.2.4 Scalability considerations for longer qubit chains 87 6.2.5 Influence of imperfect pulses 89 6.3 Designing interaction strengths for state transfer 89 6.3.1 Narrowing the search space 92 6.3.2 Constructing solutions 93 6.3.3 Numerical results 94 6.4 Extending the state transfer protocol to qudit chains 95 6.4.1 Perfect state transfer Hamiltonian for qudits 95 6.4.2 Designing interaction strengths on the qudit chain 96 6.4.3 Numerical simulation 97 7 decoupling the com motion in matter-field in- teractions ... 99 7.1 Matter-field entanglement with centre-of-mass motion 100 contents ix 7.1.1 Model Hamiltonian for the qubit-field interac- tion 102 7.1.2 Dressing