The Permanent-of-Gaussians Conjecture

Prove that approximating the permanent of most n×n matrices of independent complex Gaussians is #P-hard, as BosonSampling hardness requires.

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	1	+# The Permanent-of-Gaussians Conjecture
	2	+
	3	+In BosonSampling, photons pass through a random linear-optical network, and each output pattern occurs with probability given by the squared permanent of a submatrix that is close to a matrix of independent Gaussians. Aaronson and Arkhipov showed that a fast classical simulation of this experiment would collapse the polynomial hierarchy, provided that approximating such permanents is #P-hard for most Gaussian matrices. That missing step is the Permanent-of-Gaussians Conjecture.
	4	+
	5	+## Problem
	6	+
	7	+Let $X$ be an $n\times n$ matrix of independent standard complex Gaussian entries, each with density $\pi^{-1}e^{-\|z\|^2}$, and let
	8	+
	9	+$$\operatorname{Per}X=\sum_{\sigma\in S_n}\prod_{j=1}^{n}X_{j,\sigma(j)} .$$
	10	+
	11	+Gaussian Permanent Estimation, $\mathrm{GPE}_\times$, asks: given $X$ and $\varepsilon,\delta>0$, output $z$ with $\|z-\operatorname{Per}X\|\le\varepsilon\,\|\operatorname{Per}X\|$, with probability at least $1-\delta$ over $X$, in time $\mathrm{poly}(n,1/\varepsilon,1/\delta)$.
	12	+
	13	+Conjecture. $\mathrm{GPE}_\times$ is $\#\mathsf{P}$-hard: if an oracle $\mathcal{O}$ solves it, then $\mathsf{P}^{\#\mathsf{P}}\subseteq\mathsf{BPP}^{\mathcal{O}}$.
	14	+
	15	+The additive version $\|\mathrm{GPE}\|^2_\pm$ asks instead for $\|\operatorname{Per}X\|^2$ to within $\pm\varepsilon\, n!$, under the same conditions [1].
	16	+
	17	+## Background
	18	+
	19	+Aaronson and Arkhipov stated the conjecture in 2010 as the main complexity-theoretic challenge of their work on linear optics [1]. They proved that a classical algorithm sampling approximately from the BosonSampling distribution in polynomial time would put $\|\mathrm{GPE}\|^2_\pm$ in $\mathsf{BPP}^{\mathsf{NP}}$. Under a second conjecture, permanent anticoncentration ($\|\operatorname{Per}X\|\ge\sqrt{n!}/\mathrm{poly}(n,1/\delta)$ except with probability $\delta$), the two versions are polynomial-time equivalent, and the conjecture above would then give $\mathsf{P}^{\#\mathsf{P}}=\mathsf{BPP}^{\mathsf{NP}}$, collapsing the polynomial hierarchy to its third level [1]. This argument underlies the complexity-theoretic case for quantum advantage in BosonSampling experiments [1, 3].
	20	+
	21	+## What is known
	22	+
	23	+- Exact computation is hard on average: by extending Lipton's random self-reducibility of the permanent, computing $\operatorname{Per}X$ exactly for a $3/4+1/\mathrm{poly}(n)$ fraction of Gaussian matrices is $\#\mathsf{P}$-hard, and so is estimating $\|\operatorname{Per}X\|^2$ to within exponentially small additive error [1]. Aaronson and Arkhipov expected that $3/4$ could be lowered to $1/2+1/\mathrm{poly}(n)$ or $1/\mathrm{poly}(n)$, as over finite fields, and left this for future work [1].
	24	+- Error tolerance, measured as additive error on the output probabilities of random $n$-photon experiments: $\#\mathsf{P}$-hardness was shown for error $e^{-O(n^4)}$ by Aaronson and Arkhipov [1, 3], then $e^{-6n\log n-O(n)}$ by Bouland, Fefferman, Landau and Liu [2], and in 2025 $e^{-n\log n-n-O(n^\delta)}$ for every $\delta>0$ by Bouland, Datta, Fefferman and Hernández, under $\mathsf{BPP}^{\mathsf{NP}}$ reductions and assuming anticoncentration [3]. The conjecture corresponds to error $e^{-n\log n-n-O(\log n)}$ [3].
	25	+- A July 2026 preprint by Koehler and Leung reports a proof of the permanent anticoncentration conjecture of [1] for complex Gaussian matrices, with analogues for real and quaternionic ones; it has not yet been refereed [4]. If it holds, then by [1] the additive and multiplicative versions are equivalent, and the anticoncentration assumption in [3] is no longer needed; it does not by itself give $\#\mathsf{P}$-hardness.
	26	+
	27	+## What would count as progress
	28	+
	29	+- Improving the $O(n^\delta)$ term in the exponent of [3] to $O(\log n)$, which [3] identifies as the remaining step toward the conjecture.
	30	+- An independent check, or a simpler proof, of the anticoncentration theorem of [4], on which the equivalence of the two versions now rests.
	31	+
	32	+## References
	33	+
	34	+1. S. Aaronson and A. Arkhipov, "The computational complexity of linear optics", Theory of Computing 9(4), 143–252 (2013). [doi:10.4086/toc.2013.v009a004](https://doi.org/10.4086/toc.2013.v009a004)
	35	+2. A. Bouland, B. Fefferman, Z. Landau and Y. Liu, "Noise and the frontier of quantum supremacy", in 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), 1308–1317 (2022). [doi:10.1109/FOCS52979.2021.00127](https://doi.org/10.1109/FOCS52979.2021.00127)
	36	+3. A. Bouland, I. Datta, B. Fefferman and F. Hernández, "Exponential improvements to the average-case hardness of BosonSampling", in 2025 IEEE 66th Annual Symposium on Foundations of Computer Science (FOCS), 912–933 (2025). [doi:10.1109/FOCS63196.2025.00047](https://doi.org/10.1109/FOCS63196.2025.00047)
	37	+4. F. Koehler and P. K. Leung, "Anticoncentration of the permanent in Ginibre ensembles", preprint (2026). [arXiv:2607.20329](https://arxiv.org/abs/2607.20329)
	38	+
	39	+## Source
	40	+
	41	+Imported from [QIQCOP Zoo op_af68b033a88ef934](https://qiqc-op.com/problem/op_af68b033a88ef934/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.