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

2026-04-01

AI Summary

This work investigates the efficient approximate computation of the permanent of random matrices with a small non-zero mean, aiming to delineate the classical computational complexity boundary of BosonSampling. By analyzing the zero distribution of the random polynomial $\mathrm{per}(zJ + W)$ and leveraging Barvinok’s interpolation method, the authors achieve an efficient approximation when a sufficiently large zero-free region exists. They establish, for the first time, that when $W$ is a standard complex Gaussian matrix, all zeros lie within a disk of radius $\widetilde{O}(n^{-1/3})$, with typical zeros having magnitude $\Theta(n^{-1/2})$, thereby aligning with the average-case hardness conjecture. The result is extended to subexponential-distributed matrices and hard-core models, yielding a universal zero-free region. Consequently, the lower bound on the bias permitting efficient approximation is relaxed from $1/\mathrm{polylog}(n)$ to $\widetilde{\Omega}(n^{-1/3})$.

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 $\tildeΩ(n^{-1/3})$. 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 $(1 - ε)n$ of them, have magnitude $Θ(n^{-1/2})$. 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.

Problem

• permanent
• random matrix
• zero-free region
• boson sampling
• interpolation

Innovation

• permanent approximation
• zero-free region
• Barvinok's interpolation
• random matrix
• universality