Sampling of globally depolarized random quantum circuit

Abstract

The recent paper [F. Arute et al. Nature {\bf 574}, 505 (2019)] considered exact classical sampling of the output probability distribution of the globally depolarized random quantum circuit. In this paper, we show three results. First, we consider the case when the fidelity $F$ is constant. We show that if the distribution is classically sampled in polynomial time within a constant multiplicative error, then ${\rm BQP}\subseteq{\rm SBP}$, which means that BQP is in the second level of the polynomial-time hierarchy. We next show that for any $F\le1/2$, the distribution is classically trivially sampled by the uniform distribution within the multiplicative error $F2^{n+2}$, where $n$ is the number of qubits. We finally show that for any $F$, the distribution is classically trivially sampled by the uniform distribution within the additive error $2F$. These last two results show that if we consider realistic cases, both $F\sim2^{-m}$ and $m\gg n$, or at least $F\sim2^{-m}$, where $m$ is the number of gates, quantum supremacy does not exist for approximate sampling even with the exponentially-small errors. We also argue that if $F\sim2^{-m}$ and $m\gg n$, the standard approach will not work to show quantum supremacy even for exact sampling.