The Power of Quantum Fourier Sampling

TL;DR

This paper demonstrates that quantum Fourier sampling can generate distributions that are hard to classically approximate, under certain conjectures, thus highlighting the computational advantage of quantum over classical sampling methods.

Contribution

It introduces a new class of quantumly sampleable distributions based on efficiently specifiable polynomials, including the Permanent and Hamiltonian Cycle polynomial, and weakens the conjectures needed for hardness results.

Findings

Quantum Fourier Sampling produces distributions hard to classically approximate.

Includes a broad class of #P-hard polynomials like Permanent and Hamiltonian Cycle.

Weakens the conjectures required for quantum sampling hardness.

Abstract

A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and Shepherd, shows that quantum computers can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov take this further by considering a distribution that can be sampled efficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled classically within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson and Arkhipov conjectures. In particular, we show a general class of quantumly sampleable distributions each of which is based on an "Efficiently Specifiable" polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials. Although our construction, unlike that proposed by Aaronson and Arkhipov, likely requires a universal quantum computer, we are able to use this additional power to weaken the conjectures needed to prove approximate sampling hardness results.