Approximating the Permanent of a Random Matrix with Polynomially Small Mean: Zeros and Universality

TL;DR

This paper develops an approximation algorithm for the permanent of certain random matrices by analyzing the zero distribution of related polynomials, revealing new bounds and universality results.

Contribution

It establishes zero-free regions for the permanent polynomial of random matrices, enabling approximation algorithms at smaller biases than previously possible.

Findings

Zeros of the polynomial lie within a radius of O(n^{-1/3}) for Gaussian entries.

Most zeros have magnitude rac{1}{2})n, preventing contradictions with hardness conjectures.

Universality results for zero-free regions extend to matrices with subexponential entries.

Abstract

We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero. This question is motivated by the goal of understanding the classical complexity of linear optics and \emph{boson sampling} (Aaronson and Arkhipov '11; Eldar and Mehraban '17). Barvinok's interpolation method enables efficient approximation of the permanent, provided one can establish a sufficiently large zero-free region for the polynomial $\mathrm{per}(zJ + W)$, where $J$ is the all-ones matrix and $W$ is a random matrix with independent mean-zero entries. We show that when the entries of $W$ are standard complex Gaussians, all zeros of the random polynomial $\mathrm{per}(zJ + W)$ lie within a disk of radius $\tilde{O}(n^{-1/3})$, which yields an approximation algorithm when the bias of the entries is $\tilde{\Omega}(n^{-1/3})$. Previously, there were no efficient algorithms at biases smaller than $1/\mathrm{polylog}(n)$, and it was unknown whether there typically exist zeros $z$ with $|z| \ge 1$. As a complementary result, we show that the bulk of the zeros, namely $(1 - \epsilon)n$ of them, have magnitude $\Theta(n^{-1/2})$. This prevents our interpolation method from contradicting the conjectured average-case hardness of approximating the permanent. We also establish analogous zero-free regions for the hardcore model on general graphs with complex vertex fugacities. In addition, we prove universality results establishing zero-free regions for random matrices $W$ with i.i.d. subexponential entries.