Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise

Pedro Figueroa-Romero1, Kavan Modi2,3, and Min-Hsiu Hsieh1

1Hon Hai Quantum Computing Research Center, Taipei, Taiwan
2School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia
3Centre for Quantum Technology, Transport for New South Wales, Sydney, NSW 2000, Australia

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The rapid progress in the development of quantum devices is in large part due to the availability of a wide range of characterization techniques allowing to probe, test and adjust them. Nevertheless, these methods often make use of approximations that hold in rather simplistic circumstances. In particular, assuming that error mechanisms stay constant in time and have no dependence in the past, is something that will be impossible to do as quantum processors continue scaling up in depth and size. We establish a theoretical framework for the Randomized Benchmarking protocol encompassing temporally-correlated, so-called non-Markovian noise, at the gate level, for any gate set belonging to a wide class of finite groups. We obtain a general expression for the Average Sequence Fidelity (ASF) and propose a way to obtain average gate fidelities of full non-Markovian noise processes. Moreover, we obtain conditions that are fulfilled when an ASF displays authentic non-Markovian deviations. Finally, we show that even though gate-dependence does not translate into a perturbative term within the ASF, as in the Markovian case, the non-Markovian sequence fidelity nevertheless remains stable under small gate-dependent perturbations.

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[5] Markus Heinrich, Martin Kliesch, and Ingo Roth, "Randomized benchmarking with random quantum circuits", arXiv:2212.06181, (2022).

[6] Shih-Xian Yang, Pedro Figueroa-Romero, and Min-Hsiu Hsieh, "Machine Learning of Average Non-Markovianity from Randomized Benchmarking", arXiv:2207.01542, (2022).

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