Approximating the Permanent of a Random Matrix with Vanishing Mean

TL;DR

This paper presents a quasi-polynomial time algorithm for approximating the permanent of certain random matrices with vanishing mean, challenging the belief that the permanent is computationally hard due to sign interference.

Contribution

It introduces an efficient algorithm for approximating the permanent of random matrices with small mean, including Gaussian and Bernoulli types, in cases previously thought to be hard.

Findings

Algorithm works for matrices with mean 1/lnln(n)^{1/8}

Approximation in 2^{O(n^{ ext{ extbeta}})} time for mean 1/poly(ln(n))

Challenges the sign problem as the main obstacle in permanent computation

Abstract

We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm.