Quantum Simulation with Superconducting Circuits by Vinay Ramasesh A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Irfan Siddiqi, Chair Professor Birgitta Whaley Professor Norman Yao Summer 2019 Quantum Simulation with Superconducting Circuits Copyright 2019 by Vinay Ramasesh 1 Abstract Quantum Simulation with Superconducting Circuits by Vinay Ramasesh Doctor of Philosophy in Physics University of California, Berkeley Professor Irfan Siddiqi, Chair Nonlinear superconducting circuits are a leading candidate to implement quantum infor- mation tasks beyond the reach of classical processors. One such task, quantum simulation, involves using a controllable quantum system to study the dynamics of another. In this thesis, we present a series of experiments using small systems of superconducting circuits to perform various quantum tasks relevant to quantum simulation. In the first experiment, we design and build a circuit comprising three transmon qubits whose dynamics mirror those of interacting bosonic particles on a lattice, described by the Bose-Hubbard Hamiltonian. We verify the predictions of the Bose-Hubbard model for this system spectroscopically and then use time-domain measurements to study the decoherence processes affecting the qubits. Using a Raman process, we engineer artificial decay dynamics in the system which allow us to prepare and stabilize otherwise inaccessible states of the transmon array. The second experimental demonstration concerns topological quantum matter. Certain quantum systems are characterized by quantities known as topological invariants, which are robust to local perturbations. These topological invariants are challenging to measure in naturally-occurring systems. We engineer an artificial system, comprising a transmon qubit coupled to a high-Q cavity, capable of undergoing a protocol known as the quantum walk, which also exhibits topological invariants. By using particular non-classical state of the cavity, we modify the quantum walk protocol to make the topological invariant directly accessible. Finally, we implement a hybrid quantum-classical algorithm|known as the variational quantum eigensolver|in a two-transmon quantum processor. As a proof of principle demon- stration, we compute the ground-state and low-lying excited state energies of the hydrogen molecule as a function of nuclear separation. We show that an extension of the basic algo- rithm, known as the quantum subspace expansion, allows for the mitigation of errors caused by decoherence processes affecting the quantum processor. i To my parents, Nalini and Ranga Ramasesh, and my grandmother, Lakshmi Ramachandran, for their constant support and encouragement. ii Contents Contents ii List of Figures iv List of Tables v 1 Introduction 1 1.1 Structure of Thesis . 3 1.2 Summary of Key Results . 4 2 Superconducting Circuits: Background 6 2.1 Quantum behavior in superconducting circuits . 6 2.2 Coupling a Transmon to a Linear Cavity: the Jaynes-Cummings Hamiltonian 10 3 Cooling in a Bose-Hubbard Chain 19 3.1 Realizing the Bose-Hubbard Hamiltonian with Transmon Qubits . 20 3.2 Chip design . 22 3.3 Initial spectroscopic calibrations . 25 3.4 Natural decay dynamics . 29 3.5 Cooling via an engineered bath . 35 3.6 Conclusion . 39 4 Observing Topological Invariants: Theory 40 4.1 Background: topological features of quantum walks . 40 4.2 Measuring the winding number . 45 4.3 Experimental proposal . 52 4.4 Conclusion and Future Work . 53 5 Observing Topological Invariants: Experiment 54 5.1 Motivation . 54 5.2 Realizing quantum walks in circuit QED . 56 5.3 Details of the experimental toolbox . 62 5.4 Theoretical details . 72 iii 5.5 Conclusion . 74 6 Molecular Spectra with a Hybrid Algorithm 78 6.1 Variational Quantum Eigensolver: Theory and Background . 79 6.2 Experimental Methods . 83 6.3 Results . 85 6.4 Conclusion . 93 7 Future directions 95 Bibliography 97 iv List of Figures 2.1 Circuit QED architecture for a transmon qubit . 11 2.2 Energy levels of the Jaynes-Cummings Hamiltonian . 16 3.1 Experimental setup and spectroscopy of the Bose-Hubbard array . 23 3.2 Layout of the three-transmon chip used in the Bose-Hubbard experiment . 24 3.3 Raw spectroscopy data of the three-transmon chip . 27 3.4 Bright and dark features of the excited manifold . 29 3.5 Readout calibration histograms for the ten lowest-lying Bose-Hubbard eigenstates 31 3.6 Natural decay dynamics of the Bose-Hubbard array . 33 3.7 Engineered decay dynamics of the Bose-Hubbard array . 36 3.8 Stabilizing eigenstates of the Bose-Hubbard array against particle loss . 38 5.1 Quantum walk implementation in cavity phase space . 55 5.2 Quantum walk protocol and resulting populations . 57 5.3 Topological classes of split-step quantum walks . 59 5.4 Winding number measurement via direct Wigner tomography . 61 5.5 Quantum walk with angles θ1 = 0:28π and θ2 = 0:64π . 63 5.6 Pulse shaping for Q and Wigner Tomography . 65 5.7 Pulse shaping for the quantum walk . 69 5.8 Aluminum cavity and superconducting transmon qubit . 71 5.9 Detailed block diagram of the quantum walk measurement setup . 77 6.1 Variational quantum eigensolver algorithm and quantum subspace expansion . 80 6.2 Measurement setup used in the variational quantum eigensolver experiment . 85 6.3 VQE control parameter convergence during optimization . 86 6.4 Computed H2 energy spectrum as a function of internuclear distance . 87 6.5 Error mitigation via the quantum subspace expansion . 88 6.6 Quantum subspace expansion with various measurement operators . 89 6.7 Convergence of VQE optimizer: comparison with numerics . 92 6.8 Quantum subspace expansion beyond linear response . 93 v List of Tables 3.1 Theoretical and experimental dispersive shifts of the linear cavity coupled to the Bose-Hubbard array . 31 3.2 Experimentally-measured natural decay times (downward transitions) of the three- transmon eigenstates . 32 3.3 Experimentally-measured natural decay times (upward transitions) of the three- transmon eigenstates . 34 3.4 Theoretically-calculated, Purcell-limited natural decay times (downward transi- tions) of the three-transmon eigenstates . 35 vi Acknowledgments I am extremely grateful to have spent the last six years at Berkeley working in the Quantum Nanoelectronics Lab. It is a pleasure to thank the many people who contributed to making my time here an enjoyable and productive learning experience. Thanks first to my advisor, Dr. Irfan Siddiqi, for welcoming me into his lab and main- taining such a great environment for nurturing scientific curiosity. Irfan constantly pushes his students to perform at the best of their ability while giving them the freedom to explore their individual interests, for which I am truly grateful. My ability to think independently as a scientist and generate interesting research directions on my own has come about through his guidance, and I am very fortunate to have had such an inspiring mentor. One of our closest collaborators during my time here at Berkeley was Dr. Norman Yao, who was a postdoc when I started grad school and is now a professor. Norman was a fantastic collaborator; his knowledge of quantum physics is unparalleled and helped us to get the quantum walk experiment working. I had a great time working with him, and also discussing physics and life over tennis matches, Chengdu dinners, Marvel movie nights, and many many paper draft revisions. I have also been fortunate to work with five incredibly talented postdocs, who have shaped my skills and thinking a great deal. Dr. Shay Hacohen-Gourgy was a pleasure to work with as a new graduate student in QNL; he taught me the basics of working with dilution fridges, patiently led me through fabricating my first samples, and showed me what it means to be an experimental physicist. Dr. Emmanuel Flurin mentored me through the quantum walk project; his incredibly deep knowledge of circuit QED was a huge asset to that project, and I hope to have picked up some of his knowledge along the way. Dr. Kevin O'Brien showed me the value of carefully attacking a complex problem systematically and scientifically to produce almost magical-seeming results. I enjoyed all of our discussions through many hours of fabrication and testing, and especially appreciated his ability to hone in and focus on the most important parts of any given problem. From Dr. James Colless I learned the importance of precisely and quantitatively understanding as much of the experimental apparatus as possible. I was also constantly impressed by how hard he worked. Dr. Machiel Blok was my partner in the qutrit black hole experiment. I greatly enjoyed working with him, and learned a lot from his style of doing physics, specifically his approach to data analysis and his resilience in the face of inevitable setbacks. I wish we had still another year to pursue all the ideas we discussed over the last couple years. The entirety of QNL, past and present, deserves many thanks for making the lab a great place to work. These include graduate students, undergrads, visitors, and postdocs. I enjoyed our house of curries lunches, Tahoe trips, foosball matches, Bring Sally Up com- petition, and last but certainly not least, the invention of Cracketball. I am privileged to have worked with all of you: Eli Levenson-Falk, Chris Macklin, Mollie Schwartz, Andrew Eddins, Steven Weber, David Toyli, Nick Frattini, Aditya Venkatramani, Eunsong Kim, Al- lison Dove, Leigh Martin, John Mark Kreikebaum, Will Livingston, Brad Mitchell, Marie vii Lu, Trevor Christolini, Akel Hashim, Ravi Naik, Archan Banerjee, Dar Dahlen, Jack Qiu, and Noah Stevenson. I am very thankful for the many friends in the area who made daily life in Berkeley such a joy. Berkeley is a very tough place to leave thanks to all of you.