Boson-Sampling in the light of sample complexity

TL;DR

This paper demonstrates that, with high probability, no symmetric classical algorithm can distinguish Boson-Sampling distributions from uniform distributions with fewer than exponentially many samples, highlighting the problem's inherent classical hardness.

Contribution

It proves that symmetric algorithms cannot efficiently differentiate Boson-Sampling distributions from uniform ones, emphasizing the classical computational difficulty of verifying quantum devices.

Findings

Symmetric algorithms require exponentially many samples to distinguish distributions.

Boson-Sampling distributions are operationally indistinguishable from uniform distributions.

Efficient classical certification of Boson-Sampling devices is likely infeasible.

Abstract

Boson-Sampling is a classically computationally hard problem that can - in principle - be efficiently solved with quantum linear optical networks. Very recently, a rush of experimental activity has ignited with the aim of developing such devices as feasible instances of quantum simulators. Even approximate Boson-Sampling is believed to be hard with high probability if the unitary describing the optical network is drawn from the Haar measure. In this work we show that in this setup, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform one from fewer than exponentially many samples. This means that the two distributions are operationally indistinguishable without detailed a priori knowledge. We carefully discuss the prospects of efficiently using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this. We conclude that due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach.