Achieving quantum supremacy with sparse and noisy commuting quantum computations

TL;DR

This paper explores the computational power of sparse and noisy IQP quantum circuits, demonstrating their potential for quantum supremacy and analyzing the impact of noise and error correction on classical simulability.

Contribution

It introduces a family of sparse IQP circuits implementable on a lattice that are likely hard to simulate classically, even with noise and error correction.

Findings

Sparse IQP circuits can be implemented efficiently on a lattice.

Small noise levels allow efficient classical simulation of IQP output distributions.

Error correction can preserve the classical hardness of simulating noisy IQP circuits.

Abstract

The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.