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Tight Bounds on the Convergence of Noisy Random Circuits to the Uniform Distribution



Michael Gullans, Abhinav Deshpande, Bill Fefferman, Alexey Gorshkov, Pradeep Niroula, Oles Shtanko


We study the properties of output distributions of noisy, random circuits. We obtain upper and lower bounds on the expected distance of the output distribution from the uniform distribution. These bounds are tight with respect to the dependence on circuit depth. Our proof techniques also allow us to make statements about the presence or absence of anticoncentration for both noisy and noiseless circuits. We uncover a number of interesting consequences for hardness proofs of sampling schemes that aim to show a quantum computational advantage over classical computation. Specifically, we discuss recent barrier results for depth-agnostic and/or noise-agnostic proof techniques. We show that in certain depth regimes, noise-agnostic proof techniques might still work in order to prove an often-conjectured claim in the literature on quantum computational advantage, contrary to what was thought prior to this work.
PRX Quantum


Quantum information science, noisy quantum devices, computational complexity theory


Gullans, M. , Deshpande, A. , Fefferman, B. , Gorshkov, A. , Niroula, P. and Shtanko, O. (2022), Tight Bounds on the Convergence of Noisy Random Circuits to the Uniform Distribution, PRX Quantum, [online],, (Accessed April 15, 2024)
Created December 16, 2022, Updated January 9, 2023