Existence of a polynomial-time classical approximate sampler for boson sampling (Haar-unitary setting)

Determine whether there exists a polynomial-time classical algorithm that approximately samples from the boson sampling distribution over occupancy vectors z with sum n, where the probability of z is proportional to the square of the permanent of A_z divided by the product of z_i! and A is taken to be the first n columns of an m×m Haar-random unitary matrix.

Background

Boson sampling, as introduced by Aaronson and Arkhipov, considers sampling from a distribution whose probabilities are determined by squared permanents of submatrices derived from an m×n matrix A, typically formed by the first n columns of an m×m Haar-random unitary matrix. This distribution is believed to be hard to sample classically in polynomial time.

The paper notes known exponential-time classical simulation methods and references a widely held conjecture that no polynomial-time classical approximate sampler exists for this distribution in the standard setting. While the authors present polynomial-time classical sampling algorithms for related distributions (including a GBS-related graph distribution and a boson sampling–like distribution under non-negativity constraints on A), the general Haar-unitary case remains conjecturally hard and unresolved.