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Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits

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Performance Evaluation Methodologies and Tools (VALUETOOLS 2021)

Abstract

We study a classical model for the accumulation of errors in multi-qubit quantum computations. By modeling the error process in a quantum computation using two coupled Markov chains, we are able to capture a weak form of time-dependency between errors in the past and future. By subsequently using techniques from the field of discrete probability theory, we calculate the probability that error quantities such as the fidelity and trace distance exceed a threshold analytically. The formulae cover fairly generic error distributions, cover multi-qubit scenarios, and are applicable to the randomized benchmarking protocol. To combat the numerical challenge that may occur when evaluating our expressions, we additionally provide an analytical bound on the error probabilities that is of lower numerical complexity. Besides this, we study a model describing continuous errors accumulating in a single qubit. Finally, taking inspiration from the field of operations research, we illustrate how our expressions can be used to decide how many gates one can apply before too many errors accumulate with high probability, and how one can lower the rate of error accumulation in existing circuits through simulated annealing.

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Acknowledgments

We are grateful to Bart van Schooten, who contributed the code on TU/e’s GitLab server. Finally, this research received financial support from the Chinese Scholarship Council (CSC) in the form of a CSC Scholarship.

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, Delft, Netherlands

    Long Ma

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands

    Jaron Sanders

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  1. Long Ma
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Correspondence to Long Ma .

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Editors and Affiliations

  1. Tsinghua University, Beijing, China

    Qianchuan Zhao

  2. Sun Yat-sen University, Guangzhou, China

    Li Xia

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Ma, L., Sanders, J. (2021). Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. In: Zhao, Q., Xia, L. (eds) Performance Evaluation Methodologies and Tools. VALUETOOLS 2021. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 404. Springer, Cham. https://doi.org/10.1007/978-3-030-92511-6_3

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  • DOI: https://doi.org/10.1007/978-3-030-92511-6_3

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  • Online ISBN: 978-3-030-92511-6

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