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

• URL: http://arxiv.org/abs/2604.01367v1
• Date: Wed, 01 Apr 2026 20:23:28 GMT
• Title: Approximating the Permanent of a Random Matrix with Polynomially Small Mean: Zeros and Universality
• Abstract summary: We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero.<n>We establish a sufficiently large zero-free region for the $mathrmper(zJ + W)$, where $J$ is the all-ones matrix and $W$ is a random matrix with independent mean-zero entries.<n>We show that when the entries of $W$ are complex, all zeros of the random $mathrmper(zJ + W)$ lie within a disk of radius $tildeO(n
• Score: 11.09360259927697
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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