Average-case complexity versus approximate simulation of commuting quantum computations

TL;DR

This paper investigates the classical simulation difficulty of IQP quantum computations, linking it to average-case hardness conjectures related to statistical physics and polynomial zeroes, suggesting quantum advantage under certain assumptions.

Contribution

It introduces new average-case hardness conjectures for IQP computations based on problems in statistical physics and polynomial analysis, strengthening the case for quantum computational supremacy.

Findings

IQP computations are hard to simulate classically if certain average-case conjectures hold.

The conjectures are supported by worst-case complexity evidence.

Spin-based generalisations of Boson Sampling are introduced to avoid permanent anticoncentration issues.

Abstract

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.