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Supercomputer Simulations of Transmon Quantum Computers Supercomputer simulations of transmon quantum computers Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Dennis Willsch, M.Sc. aus Koln¨ arXiv:2008.13490v1 [quant-ph] 31 Aug 2020 Berichter: Prof. Dr. Kristel Michielsen Prof. Dr. David DiVincenzo Tag der m¨undlichen Pr¨ufung:7. Juli 2020 Diese Dissertation ist auf den Internetseiten der Universit¨atsbibliothekverf¨ugbar. Abstract We develop a simulator for quantum computers composed of superconducting transmon qubits. The simulation model supports an arbitrary number of transmons and resonators. Quantum gates are implemented by time-dependent pulses. Nontrivial effects such as crosstalk, leakage to non-computational states, entanglement between transmons and res- onators, and control errors due to the pulses are inherently included. The time evolution of the quantum computer is obtained by solving the time-dependent Schr¨odingerequation. The simulation algorithm shows excellent scalability on high- performance supercomputers. We present results for the simulation of up to 16 transmons and resonators. Additionally, the model can be used to simulate environments, and we demonstrate the transition from an isolated system to an open quantum system governed by a Lindblad master equation. We also describe a procedure to extract model parameters from electromagnetic simulations or experiments. We compare simulation results to experiments on several NISQ processors of the IBM Q Experience. We find nearly perfect agreement between simulation and experiment for quantum circuits designed to probe crosstalk in transmon systems. By studying common gate metrics such as the fidelity or the diamond distance, we find that they cannot reliably predict the performance of repeated gate applications or practical quantum algorithms. As an alternative, we find that the results from two-transmon gate set tomography have an exceptional predictive power. Finally, we test a protocol from the theory of quantum error correction and fault tolerance. We find that the protocol systematically improves the performance of transmon quantum computers in the presence of characteristic control and measurement errors. III Zusammenfassung Wir entwickeln einen Simulator fur¨ Quantencomputer, die aus supraleitenden Transmon- Qubits bestehen. Das Simulationsmodell unterstutzt¨ eine beliebige Anzahl von Trans- mons und Resonatoren. Quantengatter werden durch zeitabh¨angige Pulse realisiert. Nicht- triviale Effekte wie Crosstalk, Verlust in nicht rechnerische Zust¨ande, Verschr¨ankung zwi- schen Transmons und Resonatoren sowie Steuerungsfehler verursacht durch die Pulse sind automatisch miteinbezogen. Die Zeitentwicklung des Quantencomputers wird durch L¨osung der zeitabh¨angigen Schr¨odingergleichung bestimmt. Der Simulationsalgorithmus zeigt ausgezeichnete Skalier- barkeit auf Hochleistungs-Supercomputern. Wir pr¨asentieren Ergebnisse fur¨ die Simulati- on von bis zu 16 Transmons und Resonatoren. Zus¨atzlich kann das Modell zur Simulation von Umgebungen verwendet werden. Wir demonstrieren den Ubergang¨ von einem iso- lierten System zu einem offenen Quantensystem, das von einer Lindblad-Mastergleichung bestimmt wird. Wir beschreiben außerdem ein Verfahren zur Extraktion von Modellpa- rametern aus elektromagnetischen Simulationen oder Experimenten. Wir vergleichen Simulationsergebnisse mit Experimenten auf mehreren NISQ-Prozes- soren der IBM Q Experience. Wir finden eine nahezu perfekte Ubereinstimmung¨ zwischen Simulation und Experiment fur¨ Quantenschaltungen zur Untersuchung von Crosstalk in Transmon-Systemen. Durch Untersuchung g¨angiger Gatter-Metriken wie der Fidelity oder der Diamant-Distanz finden wir, dass sie die Leistung von wiederholten Gatteranwen- dungen oder praktischen Quantenalgorithmen nicht zuverl¨assig vorhersagen k¨onnen. Als Alternative finden wir, dass die Ergebnisse einer Zwei-Transmon-Gattermengentomogra- phie eine außergew¨ohnlich gute Vorhersagekraft aufweisen. Zum Schluss testen wir ein Protokoll aus der Theorie der Quantenfehlerkorrektur und Fehlertoleranz. Wir stellen fest, dass das Protokoll systematisch die Leistung von Transmon-Quantencomputern bei charakteristischen Steuerungs- und Messfehlern verbessert. V Contents 1 Introduction 1 2 Ideal gate-based quantum computing 5 2.1 Quantum bits . .5 2.1.1 Single qubits . .6 2.1.2 Bloch sphere . .6 2.1.3 Multiple qubits . .8 2.1.4 Leakage . 11 2.2 Quantum gates . 12 2.2.1 Unitary operators . 12 2.2.2 Elementary quantum gates . 13 2.3 Quantum circuits . 14 2.4 Quantum operations . 16 2.4.1 Representations of quantum operations . 17 2.4.2 Transformations of subsystems and leakage . 19 3 Simulating superconducting transmon qubits 21 3.1 Superconducting circuits . 21 3.1.1 Quantum and classical descriptions . 22 3.1.2 LC resonator . 23 3.1.3 Josephson junction . 24 3.1.4 Cooper pair box . 25 3.2 Transmon quantum computer model . 26 3.2.1 Hamiltonian . 26 3.2.2 Choice of the basis . 27 3.3 Simulation toolkit . 31 3.3.1 Numerical algorithm: solver ..................... 32 3.3.2 Evaluation of the results: evaluator ................. 38 3.3.3 Visualization of the results: visualizer ............... 42 3.4 Definition of the model systems . 42 3.4.1 Single transmon-resonator system . 42 3.4.2 Transmon-resonator system coupled to a bath . 42 3.4.3 Two-transmon system . 44 3.4.4 Small five-transmon system . 44 3.4.5 Large five-transmon system . 44 VII Contents 3.5 Modeling electromagnetic environments . 48 3.5.1 The Foster representation of an electromagnetic environment . 50 3.5.2 Mapping to the model Hamiltonian . 52 4 Free time evolution 59 4.1 Accuracy and performance benchmarks . 60 4.1.1 Accuracy . 60 4.1.2 Performance . 63 4.2 Single transmon-resonator system . 68 4.2.1 Overview of known perturbative results . 68 4.2.2 Comparison to simulation results . 70 4.3 Transmon-resonator system coupled to a bath . 72 4.3.1 Simulation models . 73 4.3.2 Results . 75 4.3.3 Additional ways to improve the models . 79 4.4 Effective ZZ interaction for coupled transmons . 80 4.5 Conclusions . 83 5 Optimizing pulses for quantum gates 85 5.1 Single-qubit pulses . 86 5.1.1 The VZ gate . 87 5.1.2 The GD pulse . 87 5.1.3 The zero pulse . 88 5.2 Two-qubit pulses . 89 5.2.1 CNOT gates based on the CR effect . 89 5.2.2 Analysis of IX and ZX interactions . 96 5.3 Optimization of pulse parameters . 98 5.3.1 The Nelder{Mead algorithm . 99 5.3.2 Optimization results . 101 5.4 Compiling quantum circuits . 105 5.5 Alternative gate optimization techniques . 107 5.6 Conclusions . 107 6 Errors in quantum gates 109 6.1 Evaluation of gate metrics . 110 6.1.1 Average gate fidelity . 110 6.1.2 Diamond distance . 111 6.1.3 Unitarity . 115 6.1.4 Results . 116 6.2 Repeated gate applications . 120 6.2.1 Evolution of the diamond distance . 120 6.2.2 Relation to experiments . 123 6.3 Gate set tomography . 125 6.3.1 The idea of GST . 126 6.3.2 Running GST . 129 VIII Contents 6.3.3 Predicting repeated pulse applications . 135 6.4 Conclusions . 138 7 Selected quantum circuit experiments 141 7.1 Crosstalk experiments . 142 7.1.1 Circuit and simulation results . 142 7.1.2 Comparison with experiments on the IBM Q Experience . 145 7.2 Characterization of the singlet state . 147 7.2.1 Experiment . 148 7.2.2 Effective error model . 152 7.3 Testing quantum fault tolerance . 155 7.3.1 Fault-tolerant protocol . 156 7.3.2 Test systems and circuits . 159 7.3.3 Results . 161 7.4 Conclusions . 165 8 Discussion and conclusion 167 Appendices 171 A Visualization of quantum gate implementations . 173 B Elementary gate set used for the simulation . 176 C The reason for linear and unitary transformations in quantum theory . 178 C.1 Wigner's theorem . 178 C.2 Alternative approaches . 181 C.3 General remarks . 183 D Implementations of the four-component transformations V and V y ..... 184 E Error bounds for observables . 186 F Pulse parameters for quantum gates . 187 G Average fidelity of trace-decreasing quantum operations . 190 G.1 Preliminaries . 190 G.2 Quantum information theoretic proof . 191 G.3 Analytic proof . 192 H Diamond distance between unitary quantum operations . 194 I Proof of a diamond-distance bound for trace-decreasing operations . 196 J Gate decompositions and effective Hamiltonians . 198 J.1 The matrix logarithm . 199 J.2 Extracting the Hamiltonian . 199 Bibliography 203 List of publications 231 Eidesstattliche Erkl¨arung 233 Acknowledgments 235 IX Chapter 1 Introduction For over a century, humans have designed and built digital computing machines. The initial ideas can be traced back to the mid-1800s [Bab1837; Boo1847], but the actual construction started less than a century ago. In 1936, Zuse designed a floating point general-purpose computer [Zus1936] that led to the first programmable floating point machine in 1941, the Z3 [Cop2017]. Turing formalized the universal computing machine [Tur1937] that influenced the construction of the Colossus in 1943 [Ran1973], which was used to perform Boolean operations for cryptanalysis. Other computers of that time were the ABC [Ata1940] built in 1942 and the ENIAC built in 1945 [Ran1973]. Most of these early computers.
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