The complexity of simulating constant-depth BosonSampling

TL;DR

This paper proves that simulating constant-depth BosonSampling remains classically hard unless a major complexity-theoretic collapse occurs, extending known results from quantum circuits to linear-optical networks.

Contribution

It demonstrates classical hardness for exact BosonSampling with constant-depth linear-optical networks, a significant extension of prior complexity results.

Findings

Constant-depth BosonSampling is classically hard unless the polynomial hierarchy collapses.

The result applies to networks with more than four beam splitter layers.

Extends complexity results from quantum circuits to linear-optical models.

Abstract

BosonSampling is a restricted model of quantum computation proposed recently, where a non-adaptive linear-optical network is used to solve a sampling problem that seems to be hard for classical computers. Here we show that, even if the linear-optical network has a constant number (greater than four) of beam splitter layers, the exact version of the BosonSampling problem is still classically hard, unless the polynomial hierarchy collapses to its third level. This is based on similar result known for constant-depth quantum circuits and circuits of 2-local commuting gates (IQP).