Linköping University | Department of Physics, Chemistry and Biology Master thesis, 30 ECTS | Educational Program: Physics and Nanoscience Spring term 2018 | LITH-IFM-A-EX–18/3576–SE Quantum Annealing Connuous-variable quantum annealing with superconducng circuits Pontus Vikstål Supervisor : Viktor Ivády Examiner : Igor Abrikosov External supervisor : Giulia Ferrini & Göran Johansson at Applied Quantum Physics Laboratory (AQP), Chalmers Linköpings universitet SE–581 83 Linköping +46 13 28 10 00 , www.liu.se Avdelning, institution Datum Division, Department Date 2018-08-17 Department of Physics , Chemistry and Biology Linköping University Språk Rapporttyp ISBN Language Report category Svenska/Swedish Licentiatavhandling ISRN: LITH-IFM-A-EX--18/3576--SE Engelska/English Examensarbete _________________________________________________________________ C-uppsats D-uppsats Serietitel och serienummer ISSN ________________ Övrig rapport Title of series, numbering ______________________________ _____________ URL för elektronisk version Titel Title Continuous-variable quantum annealing with superconducting circuits Författare Author Pontus Vikstål Sammanfattning Abstract Quantum annealing is expected to be a powerful generic algorithm for solving hard combinatorial optimization problems faster than classical computers. Finding the solution to a combinatorial optimization problem is equivalent to finding the ground state of an Ising Hamiltonian. In today's quantum annealers the spins of the Ising Hamiltonian are mapped to superconducting qubits. On the other hand, dissipation processes degrade the success probability of finding the solution. In this thesis we set out to explore a newly proposed architecture for a noise-resilient quantum annealer that instead maps the Ising spins to continuous variable quantum states of light encoded in the field quadratures of a two-photon pumped Kerr- nonlinear resonator based on the proposal by Puri et al. (2017). In this thesis we study the Wigner negativity for this newly proposed architecture and evaluate its performance based on the negativity of the Wigner function. We do this by determining an experimental value to when the presence of losses become too detrimental, such that the Wigner function of the quantum state during the evolution within the anneal becomes positive for all times. Furthermore, we also demonstrate the capabilities of this continuous variable quantum annealer by simulating and finding the best solution of a small instance of the NP-complete subset sum problem and of the number partitioning problem. Nyckelord Keyword quantum annealing, adiabatic quantum computing, continuous variables, coherent states, Wigner function, superconducting circuits, Kerr-nonlinear resonator. Abstract Quantum annealing is expected to be a powerful generic algorithm for solving hard com- binatorial optimization problems faster than classical computers. Finding the solution to a combinatorial optimization problem is equivalent to finding the ground state ofan Ising Hamiltonian. In today’s quantum annealers the spins of the Ising Hamiltonian are mapped to superconducting qubits. On the other hand, dissipation processes degrade the success probability of finding the solution. In this thesis we set out to explore anewly proposed architecture for a noise-resilient quantum annealer that instead maps the Ising spins to continuous variable quantum states of light encoded in the field quadratures of a two-photon pumped Kerr-nonlinear resonator based on the proposal by Puri et al. (2017). In this thesis we study the Wigner negativity for this newly proposed architecture and evaluate its performance based on the negativity of the Wigner function. We do this by determining an experimental value to when the presence of losses become too detrimen- tal, such that the Wigner function of the quantum state during the evolution within the anneal becomes positive for all times. Furthermore, we also demonstrate the capabilities of this continuous variable quantum annealer by simulating and finding the best solution of a small instance ofthe NP- complete subset sum problem and of the number partitioning problem. Keywords: quantum annealing, adiabatic quantum computing, continuous variables, coherent states, Wigner function, superconducting circuits, Kerr-nonlinear resonator. Acknowledgments First and foremost, I would like to express my deepest gratitude to my supervisor, Giulia Ferrini, for giving me this interesting project to work on. I am very grateful for the problems she has provided me with and also for her interest in this thesis. I also want to thank Göran Johansson who has kindly answered my questions regarding superconducting circuits. It has been a great privilege and opportunity to work with this highly talented team of members here at Chalmers, and I’m looking forward to stay and continue doing my research in this group. A tremendous thanks also goes out to my old high school physics teacher Andreas Hen- riksson who got me interested in pursuing a career in physics. Special thanks goes to my friends for making these past five years an enjoyable experience, in particular I would like to mention my fellow university students, Jens Roderus and Björn Hult. Finally, I would like to give my warmest thanks to Helena Liljenberg and my family. iii Contents Abstract ii Acknowledgments iii Contents iv 1 Introduction 1 1.1 Adiabatic quantum computing & quantum annealing ................ 1 1.2 Quantum annealing over continuous variables ..................... 2 1.3 Motivation & goal ..................................... 3 1.4 Organization of the thesis ................................. 3 2 Theory & background 5 2.1 Computational complexity theory ............................ 5 2.2 Ising model & qubit based quantum annealing .................... 6 2.3 Examples of combinatorial optimization problems .................. 7 2.4 Quantizing the electromagnetic field .......................... 9 2.5 Coherent states ....................................... 12 2.6 Phase space representation ................................ 15 2.7 Open quantum systems .................................. 17 2.8 Measure of closeness between quantum states ..................... 19 3 Superconducting circuits & Kerr-nonlinear resonators 21 3.1 Circuit QED ........................................ 21 3.2 Two-photon pumped Kerr-nonlinear resonator .................... 22 3.3 Two- & one-photon pumped Kerr-nonlinear resonator . 24 3.4 Coupled two-photon pumped Kerr-nonlinear resonators . 26 4 Results & discussion 29 4.1 Wigner negativity for a single Kerr-nonlinear resonator . 29 4.2 Simulation of relevant combinatorial optimization problems . 32 4.3 Remarks on scalability & the model .......................... 34 5 Conclusion 37 5.1 Future work ......................................... 37 A Appendix 39 A.1 The adiabatic theorem .................................. 39 A.2 Argument for noncommuting Hamiltonians in AQC . 41 A.3 Liouville von Neumann equation ............................ 41 A.4 Circuit Lagrangian ..................................... 42 A.5 Transformation to the rotating frame .......................... 45 A.6 Steady state & stability .................................. 46 iv A.7 Effect of single-photon pump ............................... 48 A.8 Coupling between two Kerr-nonlinear resonators ................... 48 A.9 Error estimation ...................................... 49 A.10 Generation of cat states using a two-photon pumped KNR . 50 B Code 51 B.1 Single Ising spin in a magnetic field: eigenspectrum & adiabatic condition . 51 B.2 Single Ising spin in a magnetic field: Wigner function . 52 B.3 Subset sum problem ................................... 54 B.4 Number partitioning problem .............................. 56 Bibliography 59 v 1 Introduction Quantum Annealing (QA) is a quantum algorithm that is based on Adiabatic Quantum Compu- tation (AQC), and aims at solving hard combinatorial optimization problems. A combinatorial optimization problem seeks to find the best answer to a given problem from a vast collection of configurations. A typical example of a combinatorial optimization problem is the travelling salesman problem, where a salesman seeks to find the shortest travel distance between differ- ent locations, such that all locations are visited once. The naive method to solve this problem would be to make a list of all the different routes between locations that the salesman could take and from that list find the best answer. This could work when the number of locations are few, but the naive method would fail if the number of locations grows large, since the number of possible configurations increases exponentially with the number of locations. Thus the naive method is not efficient in practice, and we should therefore develop algorithms. A combinatorial optimization problem like the travelling salesman problem can be mapped onto finding the ground state ofan Ising Hamiltonian, these problems are therefore also referred to as Ising problems. Finding the ground state of an Ising Hamiltonian can be computationally difficult [1]. The main idea of QA is to cleverly take advantage of adiabatic evolution to go from a simple non-degenerate ground state of an initial Hamiltonian to the ground state of an Ising Hamiltonian, that encodes the solution of the desired combinatorial optimization problem. The uses of QA are ample and diverse, it can for example be used for finding the low- energy conformations of lattice protein, or for finding the