加入
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.
shoal 2周前提交
0 个父提交
提交 854e0dcb35811042e4e2a11aa5f4a6f6b965664d
+41 −0 README.md 新增 查看文件
| @@ -0,0 +1,41 @@ | ||
| 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. | |