Markov chains and hitting times for error accumulation in quantum circuits

TL;DR

This paper models error accumulation in quantum circuits using coupled Markov chains, providing analytical formulas and bounds to predict error thresholds and optimize quantum gate sequences.

Contribution

It introduces a novel Markov chain framework for error modeling in quantum computations, enabling analytical calculation of error probabilities and strategies to mitigate error accumulation.

Findings

Derived analytical formulas for error probabilities in quantum circuits.

Provided bounds to efficiently evaluate error thresholds.

Demonstrated applications in optimizing gate sequences and error reduction.

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 e.g. 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 e.g. 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.