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

2026-05-20

AI Summary

This work establishes the optimal approximation ratio for computing the permanent of Hermitian positive semidefinite matrices in deterministic polynomial time. By formulating a concave optimization problem based on the row vectors of a matrix factorization, analyzing the Wick integral formula via entropy methods, and leveraging complex matrix decomposition techniques, the authors derive tight upper and lower bounds on the permanent. Their main contribution is the first complete characterization of the optimal approximation ratio at exponential precision, proving that $\operatorname{per}(A)$ satisfies $e^{-\gamma n} \widehat P(A) \le \operatorname{per}(A) \le \widehat P(A)$. This yields an approximation ratio of $e^{(\gamma+o(1))n}$, which exactly matches the known computational hardness lower bound, thereby closing a longstanding gap in the theoretical understanding of this problem.

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 \[ Φ(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{Φ(V)}. \] We prove the exact sandwich \[ e^{-γn}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] Here $γ$ is the Euler--Mascheroni constant. Since the maximization is concave, this gives a deterministic polynomial-time $e^{(γ+\varepsilon)n}$-approximation for every $\varepsilon>0$. Combined with the previous $e^{(γ-\varepsilon)n}$-hardness of approximation for positive semidefinite permanents, this resolves the optimal exponential approximation ratio for deterministic polynomial-time algorithms as $e^{(γ+o(1))n}$, 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 $γ$ per factor because $\mathbb{E}[\log T]=-γ$ 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.

Problem

• permanent
• positive semidefinite matrices
• approximation ratio
• deterministic polynomial-time
• Hermitian matrix

Innovation

• permanent approximation
• positive semidefinite matrices
• deterministic polynomial-time algorithm
• concave optimization
• entropy method