Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits

Long Ma*, Jaron Sanders

*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volumeConference contributionScientificpeer-review

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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.

Original languageEnglish
Title of host publicationPerformance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings
EditorsQianchuan Zhao, Li Xia
Number of pages20
ISBN (Print)9783030925109
Publication statusPublished - 2021
Event14th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2021 - Virtual, Online
Duration: 30 Oct 202131 Oct 2021

Publication series

NameLecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST
Volume404 LNICST
ISSN (Print)1867-8211
ISSN (Electronic)1867-822X


Conference14th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2021
CityVirtual, Online

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Error accumulation
  • Markov chains
  • Quantum circuits


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