Optimal $e^{(\gamma+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices

TL;DR

This paper establishes the optimal exponential approximation ratio for the permanent of positive semidefinite matrices within polynomial time, matching the known hardness bounds, using entropy and Wick integral techniques.

Contribution

It provides the first deterministic polynomial-time algorithm achieving the optimal exponential approximation ratio for the permanent of positive semidefinite matrices, matching the hardness bound.

Findings

Achieves a deterministic polynomial-time $e^{( ext{gamma}+ ext{epsilon})n}$-approximation.

Proves the approximation ratio is optimal under P ≠ NP.

Uses entropy arguments and Wick integral formula in the proof.

Abstract

We determine, up to lower-order terms in the exponent, the best possible deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix. If $A\succeq 0$ has no zero diagonal entry, $d=\operatorname{rank}(A)$, $A=VV^\dagger$ with $V\in\mathbb{C}^{n\times d}$ full column rank, and $v_1,\ldots,v_n$ are the rows of $V$, define \[ \Phi(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{\Phi(V)}. \] We prove the exact sandwich \[ e^{-\gamma n}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] Here $\gamma$ is the Euler--Mascheroni constant. Since the maximization is concave, this gives a deterministic polynomial-time $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon>0$. Combined with the previous $e^{(\gamma-\varepsilon)n}$-hardness of approximation for positive semidefinite permanents, this resolves the optimal exponential approximation ratio for deterministic polynomial-time algorithms as $e^{(\gamma+o(1))n}$, assuming $\mathrm{P}\ne\mathrm{NP}$. The proof is an entropy argument applied to the standard Wick integral formula for $\operatorname{per}(A)$; the loss is exactly $\gamma$ per factor because $\mathbb{E}[\log T]=-\gamma$ for $T\sim\operatorname{Exp}(1)$. The result was obtained through interactions with GPT 5.5 Pro Extended: the first author's interaction was one-shot, while the second author's was a separate multi-turn interaction with high-level guidance. Both authors verified the theorem and proof. Codex was used to assemble and typeset the manuscript.