Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Frederic Koehler, Pui Kuen Leung

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.

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

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 , where is the all-ones matrix and is a random matrix with independent mean-zero entries. We show that when the entries of are standard complex Gaussians, all zeros of the random polynomial lie within a disk of radius , which yields an approximation algorithm when the bias of the entries is . Previously, there were no efficient algorithms at biases smaller than , and it was unknown whether there typically exist zeros with . As a complementary result, we show that the bulk of the zeros, namely of them, have magnitude . 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 with i.i.d. subexponential entries.

Paper Structure

This paper contains 84 sections, 54 theorems, 757 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Previous work.
  3. Our results.
  4. Further related work
  5. Overview
  6. Riemann sphere perspective
  7. Previous approaches
  8. Algorithm of Ji, Jin, and Lu ji2021approximating.
  9. New approach
  10. Analogy between permanent and hardcore model.
  11. Cluster expansion of idealized hardcore model.
  12. Convergence and zero-free region.
  13. Gain via cancellation.
  14. Generalization to hardcore model on arbitrary graphs.
  15. Universality.
  16. ...and 69 more sections

Key Result

Theorem 1

Let $\gamma, \beta > 0$ and $\delta > 0$. There exists a deterministic algorithm such that for all sufficiently large $n \ge n_0(\gamma,\beta,\delta)$, the algorithm outputs an $n^{-\gamma}$-approximation to $\log \mathop{\mathrm{per}}\nolimits(Jz + W)$ for any $|z|\ge n^{-1/3+\beta}$ with probabili

Figures (6)

  • Figure 1: Connected diagrams contributing to the first three orders of the formal expansion of $\log Z(z)$ for the idealized monomer--dimer model on $K_{n,n}$.
  • Figure 2: Magnitude (left) and phase (right) of the function $\mathop{\mathrm{per}}\nolimits(zJ/\sqrt{n} + W)$ of a $21 \times 21$ dimensional random matrix $W$ with i.i.d. standard complex Gaussian entries. Zeros are visible as dark blue dots on the left, and by argument principle considerations on the right.
  • Figure 3: Complex magnitude of the $n \times n$ permanent \ref{['eqn:experiment-eqn']} with random complex Gaussian entries. Colors correspond to the log-magnitude of the permanent at a particular point in $\mathbb C$, with dark blue corresponding to smaller permanents (more negative log-permanent) and bright yellow to larger values (more positive log-permanent). In each case, the locations of the zeros can be visually seen as dark-blue dots in the plot.
  • Figure 4: Complex magnitude of the $n \times n$ permanent \ref{['eqn:experiment-eqn']} with random complex Laplace entries. As before, the zeros are visible as dark blue dots. The results appear qualitatively similar to the Gaussian case (Figure \ref{['fig:ryser_gaussian']}).
  • Figure 5: Complex magnitude of the $n \times n$ permanent \ref{['eqn:experiment-eqn']} with real Gaussian entries. As before, the zeros are visible as dark blue dots. Because the coefficients are real-valued, and therefore invariant under complex conjugation (which maps $i$ to $-i$), the roots exhibit symmetry across the real axis.
  • ...and 1 more figures

Theorems & Definitions (110)

  • Theorem 1: Short version of Theorem \ref{['thm:degree-accuracy-tradeoff']}
  • Remark 2: Comparison with Heilmann-Lieb theorem heilmann1972theory
  • Remark 3: Cancellations in cluster expansion
  • Lemma 4
  • proof
  • Proposition 5: Kotecký--Preiss criterion
  • Lemma 6: Lemma 3.4 of pfister1991large
  • Lemma 7
  • proof
  • Remark 8
  • ...and 100 more