Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits

Long Ma; Jaron Sanders

doi:10.1007/978-3-030-92511-6_3

Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits

Long Ma^*, Jaron Sanders

^*Corresponding author for this work

Network Architectures and Services

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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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.

Original language	English
Title of host publication	Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings
Editors	Qianchuan Zhao, Li Xia
Publisher	Springer
Pages	36-55
Number of pages	20
ISBN (Print)	9783030925109
DOIs	https://doi.org/10.1007/978-3-030-92511-6_3
Publication status	Published - 2021
Event	14th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2021 - Virtual, Online Duration: 30 Oct 2021 → 31 Oct 2021

Publication series

Name	Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST
Volume	404 LNICST
ISSN (Print)	1867-8211
ISSN (Electronic)	1867-822X

Conference

Conference	14th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2021
City	Virtual, Online
Period	30/10/21 → 31/10/21

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
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.

Keywords

Error accumulation
Markov chains
Quantum circuits

Access to Document

10.1007/978-3-030-92511-6_3

Ma_Sanders2021_Chapter_MarkovChainsAndHittingTimesForFinal published version, 603 KB

Cite this

Ma, L., & Sanders, J. (2021). Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. In Q. Zhao, & L. Xia (Eds.), Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings (pp. 36-55). (Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST; Vol. 404 LNICST). Springer. https://doi.org/10.1007/978-3-030-92511-6_3

Ma, Long ; Sanders, Jaron. / Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings. editor / Qianchuan Zhao ; Li Xia. Springer, 2021. pp. 36-55 (Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST).

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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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author = "Long Ma and Jaron Sanders",

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Ma, L & Sanders, J 2021, Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. in Q Zhao & L Xia (eds), Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings. Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST, vol. 404 LNICST, Springer, pp. 36-55, 14th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2021, Virtual, Online, 30/10/21. https://doi.org/10.1007/978-3-030-92511-6_3

Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. / Ma, Long; Sanders, Jaron.
Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings. ed. / Qianchuan Zhao; Li Xia. Springer, 2021. p. 36-55 (Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST; Vol. 404 LNICST).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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Ma L, Sanders J. Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. In Zhao Q, Xia L, editors, Performance Evaluation Methodologies and Tools - 14th EAI International Conference, VALUETOOLS 2021, Proceedings. Springer. 2021. p. 36-55. (Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, LNICST). doi: 10.1007/978-3-030-92511-6_3