Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Seth T. Merkel1, Emily J. Pritchett1, and Bryan H. Fong1,1

1HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA

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We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami [19], we provide an alternative proof of Wallman's [32] and Proctor's [28] bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.

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[1] Robin Harper and Steven T. Flammia, "Fault-Tolerant Logical Gates in the IBM Quantum Experience", Physical Review Letters 122 8, 080504 (2019).

[2] Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson, "Bounding the average gate fidelity of composite channels using the unitarity", New Journal of Physics 21 5, 053016 (2019).

[3] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia, "Robust Shadow Estimation", PRX Quantum 2 3, 030348 (2021).

[4] Jonas Helsen, Joel J. Wallman, Steven T. Flammia, and Stephanie Wehner, "Multi-qubit Randomized Benchmarking Using Few Samples", arXiv:1701.04299.

[5] Arnaud Carignan-Dugas, Kristine Boone, Joel J. Wallman, and Joseph Emerson, "From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters", New Journal of Physics 20 9, 092001 (2018).

[6] Steven T. Flammia and Joel J. Wallman, "Efficient estimation of Pauli channels", arXiv:1907.12976.

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[8] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman, "Real Randomized Benchmarking", arXiv:1801.06121.

[9] Samuele Ferracin, Theodoros Kapourniotis, and Animesh Datta, "Accrediting outputs of noisy intermediate-scale quantum computing devices", arXiv:1811.09709.

[10] Jiaan Qi and Hui Khoon Ng, "Comparing the randomized benchmarking figure with the average infidelity of a quantum gate-set", International Journal of Quantum Information 17 4, 1950031 (2019).

[11] Kristine Boone, Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson, "Randomized benchmarking under different gate sets", Physical Review A 99 3, 032329 (2019).

[12] Salonik Resch and Ulya R. Karpuzcu, "Benchmarking Quantum Computers and the Impact of Quantum Noise", arXiv:1912.00546.

[13] Martin Kliesch and Ingo Roth, "Theory of quantum system certification: a tutorial", arXiv:2010.05925.

[14] Jahan Claes, Eleanor Rieffel, and Zhihui Wang, "Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking", arXiv:2011.00007.

[15] Matthew Girling, Cristina Cirstoiu, and David Jennings, "Estimation of correlations and non-separability in quantum channels via unitarity benchmarking", arXiv:2104.04352.

[16] Jonas Helsen and Stephanie Wehner, "A benchmarking procedure for quantum networks", arXiv:2103.01165.

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