Quantum Simulation with Periodically Driven Superconducting Circuits by Mahdi Sameti Submitted in conformity with the requirements for the degree of Doctor of Philosophy in Physics School of Engineering & Physical Sciences Submitted April, 2018 The copyright in this thesis is owned by the author. Any quotation from this thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. Abstract Superconducting quantum circuits have made tremendous advances in re- alizing engineered quantum dynamics for quantum simulation and quantum information processing over the past two decades. Technological developments in the field of superconducting circuits have raised them to be the leading plat- form for implementing many-qubit systems. This thesis introduces a sequence of concepts for engineering spin-lattice Hamiltonians in analog quantum simu- lation with superconducting circuits. Our approach to quantum simulation is to engineer driving schemes that lead to implementations of the desired mod- els. The applications of this approach are in our work mainly centered around two types of systems: interesting many-body topological quantum systems, namely Kitaev's toric code and honeycomb model and two-body systems that can be employed as building blocks of larger quantum simulators or quantum computers. In the first part of this thesis, we make a proposal for an analog implementation of the toric code in superconducting circuits. We also dis- cuss a realistic implementation of this model on an eight qubit lattice. In the second part, we present our analog approach for implementing arbitrary spin- spin interactions in linearly and nonlinearly coupled superconducting qubits. Our proposed toolbox has the potential of easy generalization to a variety of systems and interactions. Lastly, based on our two-body toolbox, we set for- ward two possible implementations of the Kitaev honeycomb model and show that the engineered two-body interactions work - with good accuracy - in this many-body model. 1 Acknowledgements I would like to first thank my supervisor Prof. Michael J. Hartmann for providing the opportunity to do my Ph.D. with him. I was so lucky to have such a nice, caring and supportive person as my supervisor and I'm very much grateful to him for all his continuous supports, advices, encouragements, flexi- bility, patience and kindness during the course of my M.Sc. and Ph.D. and for all I've learnt from him. I would like to thank our collaborators in the Quan- tum Device Lab at ETH Zurich, Prof. Andreas Wallraff, Dr. Anton Poto˘cnik and Michele Collodo for contributing in the project on the toric code and for giving me the opportunity to be a part of their experimental project on a non- linear coupler. Special thanks goes to my former colleagues at TUM Munich, Dr. Mehdi Abdi, Dr. Peter Degenfeld-Schonburg and Dr. Martin Leib and to my colleagues at Heriot-Watt university, Oliver Brown, with who I shared most of my journey towards my Ph.D., and Dr. Edmund Owen. I would like to extend my warm thanks to Dr. Luca Tagliacozzo for insightful discussions on lattice gauge theories and tensor networks. Huge thanks goes to all my friends who made my Ph.D. a much better and positive experience. Sincere thanks goes to the heads of the department Prof. D. Reid and Prof. G. Buller for their supports towards my scientific travels and to Loraine Markland for all her assistance. This work is dedicated to my parents. Without your immense and precious support and attention, I never would have reached this stage. Thank you very much for all you did for me before coming abroad to study, after that in Ger- many and Scotland and for all you will do for me in the future! Thank you very much for being there whenever I needed you, raising my hopes during hopeless times, carrying the pressures with me shoulder to shoulder and tolerating my bitter attitudes. I'm well aware that I could never ever repay any of my debts to you but I hope, through this work, I could give you moments of happiness as a gift. Next to my parents, I'm deeply indebted to my sister and brother, Marsa and Mohammad, whose never-ending kindness was the best companion in my life. May each day of life bring its best for you! 2 Contents 1 Introduction 7 2 Superconducting quantum circuits 10 2.1 Operating regime of superconducting circuits . 11 2.2 Quantum LC oscillator . 11 2.3 Josephson junctions . 15 2.3.1 Josephson equations and Josephson inductance . 18 2.3.2 Tunable Josephson junction: dc-SQUID . 20 2.4 Superconducting qubits . 22 2.4.1 From Cooper pair box (CPB) to transmon . 26 2.5 Circuit Quantum Electrodynamics (cQED) . 32 2.6 Coupling schemes for transmon qubits . 36 2.7 Measurement of transmon qubit . 39 2.7.1 Joint readout of transmons . 43 2.8 Circuit network theory . 43 2.9 Summary .............................. 47 3 Quantum simulation with superconducting circuits 48 3.1 What is a quantum simulator? . 49 3.2 Digital quantum simulation . 50 3.3 Analog quantum simulation . 52 3.4 Criteria for quantum simulators . 52 3.5 Quantum simulation with superconducting circuits . 54 3 CONTENTS 3.6 Summary .............................. 57 4 Analog quantum simulation of the toric code 58 4.1 Stabilizer codes . 60 4.2 Thetoriccodemodel........................ 61 4.2.1 Excitations of the toric code . 63 4.3 Superconducting lattice for the toric code . 68 4.4 Lagrangian and Hamiltonian of the lattice . 70 4.5 Derivation of an effective Hamiltonian . 73 4.5.1 Schrieffer-Wolff transformation . 73 4.5.2 Second quantized form of the non-interacting theory . 75 4.5.3 Adiabatic elimination of the SQUID modes . 76 4.6 Engineering the time-dependent Hamiltonian . 80 4.6.1 Generation of stabilizer interactions . 81 4.6.2 Perturbations . 84 4.6.3 Perturbations in HIq .................... 86 (0) 4.6.4 Perturbations in HqS;s ................... 87 4.6.5 Corrections to adiabatic elimination: perturbations from (1) HqS ............................. 88 4.7 Towards experimental implementation . 91 4.8 Adiabatic preparation of topological order . 94 4.9 Expected measurement outcomes . 97 4.10Summary .............................. 98 5 Floquet engineering of arbitrary spin-spin interactions 100 5.1 Basic elements of Floquet theory . 101 5.1.1 Time-independent representation . 103 5.1.2 Generalization to many-mode Floquet theory . 107 5.2 Effective description of Floquet Hamiltonian . 110 5.2.1 High-frequency regime . 111 5.2.2 Resonant regime . 112 5.3 Driven qubit schemes . 117 5.3.1 Single-mode driven system . 119 5.3.2 Bimodally driven system . 129 4 CONTENTS 5.4 Driven coupling schemes . 132 5.4.1 Rotating Wave Approximation (RWA) . 134 5.4.2 Beyond RWA: Bimodal Floquet Theory . 135 5.5 Summary .............................. 137 6 Quantum simulation of Kitaev honeycomb lattice model 138 6.1 Kitaev Honeycomb lattice model . 138 6.2 Implementation based on driven qubits . 142 6.3 Implementation based on driven couplings . 149 6.4 Discussion and outlook . 151 6.5 Summary .............................. 153 7 Conclusions and future directions 154 A Expressions for Schrieffer-Wolff transformation 159 B Projection-operator approach to Salwen perturbation theory 161 Bibliography 164 5 Publication List This thesis is based in part on the following publications, Mahdi Sameti, Anton Poto˘cnik, Dan E. Browne, Andreas Wallraff, Michael J. Hartmann, Superconducting quantum simulator for topological order and the toric code, Phys. Rev. A 95, 042330 (2017). Mahdi Sameti and Michael J. Hartmann, Floquet engineering of arbitrary spin-spin interactions, to be submitted. Other publication which is not included in this thesis, M. Abdi, P. Degenfeld-Schonburg, M. Sameti, C. Navarrete-Benlloch and M. J. Hartmann, Dissipative optomechanical preparation of macroscopic quantum superposition states, Phys. Rev. Lett. 116, 233604 (2016). 6 Chapter 1 Introduction This work combines several subjects that have attracted a great deal of at- tention in recent years in the physics community: superconducting quantum circuits, quantum simulation, topological phases of matter and periodically driven quantum systems. Superconducting circuits are a solid-state platform to test the quantum optical phenomena in the microwave regime. These cir- cuits are endowed with a nonlinear element called Josephson junction which makes it feasible to define two-level quantum systems (i.e. qubits) in such circuits. Thanks to the rapid progresses in the fabrication and measurement of superconducting circuits, they have nowadays been elevated to the position of a forerunner in the race towards implementing quantum computers. Using superconducting circuits, large scale systems, primarily considered to belong to condensed matter physics, can now be studied with photons on a circuit lattice. In particular, our goal in this thesis is to exploit the potential of supercon- ducting circuits in implementing quantum simulators for intriguing many-body models, i.e. Kitaev topological models, as well as few-body Hamiltonians. The topological models are interesting on their own since they feature phenomena like degeneracy and non-locality of the ground states and exotic quasiparticles called anyons. These phases have broadened our views on phase transitions as the contemporary setting of symmetry-breaking does not apply to them and topological phase transitions do not involve symmetry-breaking. Moreover the 7 Chapter 1. Introduction nonlocality of their ground states makes them a perfect candidate to encode quantum information since local perturbations cannot destroy quantum in- formation stored in such states. In addition, one can perform fault-tolerant quantum information processing based on anyons. Unfortunately the rele- vant models do not have direct natural realizations and any implementation of them demands for an artificial quantum system. In this work, based on super- conducting circuits platform, we focus on analog implementations of Kitaev's toric code and honeycomb lattice models.