Complex-Valued-Matrix Permanents: SPA-based Approximations and Double-Cover Analysis

TL;DR

The paper addresses the challenge of approximating the permanent of complex-valued matrices, motivated by Boson sampling, by extending normal factor graph methods to double-edge DE-NFGs and analyzing SPA-based Bethe approximations. It develops DE-NFG representations for $|\operatorname{perm}(\boldsymbol{\theta})|^2$, adapts SPA to complex-valued messages, and studies double covers to understand Bethe accuracy, yielding exact and asymptotic results for key ensembles. In particular, it derives precise expressions and asymptotics for $\mathbb{E}[Z^2]$ and $\mathbb{E}[Z_{\mathrm{B},2}^2]$ under all-ones and zero-mean complex matrices, revealing regimes where the Bethe approximation remains meaningful and where it degrades as distributions become fully complex. The findings provide both algorithmic guidance for SPA on complex permanents and graph-cover analyses that illuminate the structure of Bethe approximations in complex-valued settings, with implications for Boson sampling and related probabilistic inference tasks.

Abstract

Approximating the permanent of a complex-valued matrix is a fundamental problem with applications in Boson sampling and probabilistic inference. In this paper, we extend factor-graph-based methods for approximating the permanent of non-negative-real-valued matrices that are based on running the sum-product algorithm (SPA) on standard normal factor graphs, to factor-graph-based methods for approximating the permanent of complex-valued matrices that are based on running the SPA on double-edge normal factor graphs. On the algorithmic side, we investigate the behavior of the SPA, in particular how the SPA fixed points change when transitioning from real-valued to complex-valued matrix ensembles. On the analytical side, we use graph covers to analyze the Bethe approximation of the permanent, i.e., the approximation of the permanent that is obtained with the help of the SPA. This combined algorithmic and analytical perspective provides new insight into the structure of Bethe approximations in complex-valued problems and clarifies when such approximations remain meaningful beyond the non-negative-real-valued settings.

Figures (10)

• Figure 1: NFG for $\operatorname{perm}(\boldsymbol{\theta})$ when $n=5$. Left: $\mathsf{N}(\boldsymbol{\theta})$ with edge weights $f_{ i,j }$ omitted. Right: Zoomed in on parts of $\mathsf{N}(\boldsymbol{\theta})$.
• Figure 2: DE-NFG for $|\operatorname{perm}(\boldsymbol{\theta})|^2$ when $n=5$. Left: $\mathsf{N}_{\mathrm{DE}}(\boldsymbol{\theta})$ with edge weights $\widetilde{f}_{ i,j }$ omitted. Right: Zoomed in on parts of $\mathsf{N}_{\mathrm{DE}}(\boldsymbol{\theta})$.
• Figure 3: Numerical results of SPA on DE-NFG $\mathsf{N}_{\mathrm{DE}}(\boldsymbol{\theta})$ for $n = 10$.
• Figure 4: Numerical results of double cover analysis on DE-NFG $\mathsf{N}_{\mathrm{DE}}(\boldsymbol{\theta})$ for $n=4$.
• Figure 5: A valid configuration of NFG $\widehat{N}(\boldsymbol{\theta})$. Possible $(1,1)$-edge, $(0,1)$-cycle and $(1,0)$-cycle are colored in blue, red and green, respectively.

Theorems & Definitions (34)

• Proposition 1
• proof
• Example 2
• Example 2: continued
• Definition 3
• Definition 4
• Theorem 5
• proof
• Corollary 6: of Thm. \ref{['thm:double:cover:allone']}
• proof