A walk through quantum noise: a study of error signatures and characterization methods by Arnaud CARIGNAN-DUGAS A thesis presented to the University of Waterloo in fulfilment of the thesis requirement of a Doctor of Philosophy in Applied Mathematics Waterloo, Ontario, Canada, 2019 c Arnaud Carignan-Dugas 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Prof. Jens Eisert Supervisor: Prof. Joseph Emerson Internal Member: Prof. Achim Kempf Internal-External Member: Prof. David Cory Other Member: Prof. Joel Wallman ii Author’s Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions This thesis contains literal transcriptions of articles for which my contribution was major, by which I mean that I heavily contributed to the derivation of central results. The chapters based on those articles begin with a short contribution statement and a reference to the work in question. iv Abstract The construction of large scale quantum computing devices might be one of the most ex- citing and promising endeavors of the 21st century, but it also comes with many challenges. As quantum computers are supplemented with more registers, their error profile generally grows in complexity, rendering the enterprise of quantifying the reliability of quantum com- putations increasingly difficult through naive characterization techniques. In the last decade, a lot of efforts has been directed toward developing highly scalable benchmarking schemes. A leading family of characterization methods built upon scalable principles is known as ran- domized benchmarking (RB). In this thesis, many tools are presented with the objective of improving the scalability, and versatility of RB techniques, as well as demonstrating their reliability under various error models. The first part of this work investigates the connection between the error of individual circuit components and the error of their composition. Before reasoning about intricate cir- cuit constructions, it is shown that there exists a well-motivated way to define decoherent quantum channels, and that every channel can be factorized into a unitary-decoherent com- position. This dichotomy carries to the circuit evolution of important error parameters by assuming realistic error scenarios. Those results are used to improve the confidence interval of RB diagnoses and to reconcile experimentally estimated parameters with physically and operationally meaningful quantities. In the second part of this thesis, various RB schemes are either developed or more rig- orously analyzed. A first result consists of the introduction of “dihedral benchmarking”, a technique which, if performed in conjunction with standard RB protocols, enables the char- acterization of operations that form a universal gate-set. Finally, rigorous analysis tools are provided to demonstrate the reliability of a highly scalable family of generator-based RB protocols known as direct RB. v Acknowledgements First, I would like to thank Joseph Emerson and Joel J. Wallman who both provided me with guidance during the past years. I would also like to thank Timothy J. Proctor for being an understanding and insightful research collaborator. All the rest of my acknowledgments minus one go to my family, friends and academic peers, who supported me directly or indirectly: Gigi, Lulu, Max, Tobi, Bohdan, Beenie, Bouhmi, Caro, Cary, Dan, David, Hammam, Hina, Jeremy, Joel, John, John, Jonas, JP, Katie, Lindsay, Matthew, Piers, Sasha, Tamara, Thom, Yuval, and many more. Finally, I would like to thank Margot (and her garden) who provided me support, and sometimes food. vi Table of Contents Examining Committee Membership ii Author’s Declaration iii Statement of Contributions iv Abstract v Acknowledgements vi List of Figures xi List of Tables xiv List of Abbreviations xv List of Symbols xvi 1 Introduction1 2 Introductory Material5 2.1 Inner product and norms...............................5 2.1.1 Common operator bases...........................6 2.2 Quantum states and measurements.........................7 2.3 Quantum operations.................................8 2.3.1 Definition...................................8 2.4 Mapping abstract quantum operations to more familiar sets..........8 2.4.1 Process matrix (Liouville representation).................8 2.4.2 Chi matrix................................... 11 2.4.3 Choi matrix.................................. 11 2.4.4 Canonical Kraus decomposition...................... 12 2.5 Elementary types of quantum channels...................... 12 2.5.1 Unitary errors................................. 12 2.5.2 Convexity of the set of CPTP maps..................... 14 2.5.3 Incoherent channels............................. 14 2.5.4 Stochastic errors............................... 14 2.5.5 Depolarizing errors.............................. 15 2.5.6 Dephasing/phase damping errors..................... 15 2.5.7 Amplitude damping channels (population transfer)........... 16 2.6 Quantum circuits and the Markovian assumption................ 17 vii 2.7 Characterizing quantum operations........................ 19 2.8 Reference bases and gauge transformations.................... 21 I Error signatures in quantum circuits 23 3 Bounding the average gate fidelity of composite channels using the unitarity 25 3.1 Foreword........................................ 25 3.2 Compendium..................................... 25 3.3 Introduction...................................... 26 3.4 Noisy quantum processes.............................. 27 3.5 Composite infidelities in terms of component infidelities............ 29 3.6 Improved bounds on the infidelity using the unitarity.............. 32 3.7 Application: Interleaved RB............................. 35 3.8 Summary and outlook................................ 38 3.9 Proofs.......................................... 39 3.9.1 Matrix inequalities on the real field..................... 39 3.10 Afterword....................................... 41 4 A polar decomposition for quantum channels 43 4.1 Foreword........................................ 43 4.2 Compendium..................................... 43 4.3 Introduction...................................... 43 4.4 Channel properties captured by the leading Kraus operator........... 46 4.5 The LK approximation and two evolution theorems............... 51 4.6 A polar decomposition for quantum channels................... 53 4.6.1 Defining decoherence............................ 53 4.6.2 The dynamics induced from decoherent channels as infinitesimal gen- erators..................................... 55 4.6.3 Further justifying our notion of decoherence............... 57 4.7 Behavioral signatures of coherence and decoherence............... 58 4.7.1 Extremal dephasers, extremal unitaries, and equable error channels.. 59 Extremal dephasers.............................. 59 Extremal unitaries.............................. 59 Equable error channels............................ 61 4.7.2 Reasoning about U .............................. 62 4.7.3 The coherence level.............................. 65 4.7.4 Bounding the worst and best case fidelity of a circuit.......... 66 4.7.5 Decoherence-limited operations...................... 68 4.7.6 Limitations.................................. 69 4.8 Conclusion....................................... 70 4.9 Proofs.......................................... 74 4.9.1 A noteworthy trace inequality....................... 74 4.9.2 Proofs of the main results.......................... 76 Notation and remarks............................ 76 Proof of the evolution theorem7...................... 76 Proof of the evolution theorem8...................... 77 viii Proof of theorem 12.............................. 82 Proof of theorem 11.............................. 84 Proof of theorem 13.............................. 87 Proof of theorem 10.............................. 88 Proof of theorem 14.............................. 89 Proof of theorem 15.............................. 92 4.10 Afterword....................................... 93 4.10.1 A bound on the spectral radius of channel deviations.......... 93 4.10.2 A bound on the diamond distance..................... 97 4.10.3 End of the first part.............................. 100 II Quantum characterization through randomized benchmarking 101 5 From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters 103 5.1 Foreword........................................ 103 5.2 Compendium..................................... 103 5.3 Introduction...................................... 103 5.4 The dynamics of the gate-set circuit fidelity.................... 105 5.5 Finding the appropriate set of targeted gate realizations for specific noise models109 5.6 Conclusion....................................... 111 5.7 Supplementary material............................... 112 5.7.1 An expression for the total change in the gate-set circuit fidelity.... 112 5.7.2 Varying the ideal gate-set of comparison................. 114 5.8 Afterword....................................... 120 5.8.1 Extending the results to all dimensions.................. 120 5.8.2 Comparing