Quantum Supremacy and the Complexity of Random Circuit Sampling

TL;DR

This paper provides complexity-theoretic evidence that Random Circuit Sampling (RCS), a candidate for demonstrating quantum supremacy, is both hard to simulate classically and possesses properties like anti-concentration, supporting its experimental viability.

Contribution

It proves that RCS has average-case hardness and anti-concentration, strengthening the theoretical foundation for quantum supremacy demonstrations.

Findings

RCS is #P-hard on average for typical quantum circuits.

RCS exhibits anti-concentration, ensuring output distributions are sufficiently spread.

The complexity reduction exploits the polynomial structure of quantum circuit amplitudes.

Abstract

A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS satisfies an anti-concentration property, making it the first supremacy proposal with both average-case hardness and anti-concentration.