Sampling of globally depolarized random quantum circuit

TL;DR

This paper analyzes the classical hardness of sampling from globally depolarized random quantum circuits, showing that under realistic conditions, quantum supremacy is unlikely for approximate sampling due to trivial classical algorithms.

Contribution

It provides theoretical bounds and arguments demonstrating that depolarized quantum circuits can be classically simulated efficiently in realistic regimes, challenging claims of quantum supremacy.

Findings

Classical sampling is efficient if fidelity F is constant, implying BQP in SBP.

For F ≤ 1/2, uniform distribution trivially samples the output within a certain error.

For any F, trivial classical sampling is possible within an additive error proportional to F.

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.