arXiv 1810.059072018-10math.PRcond-mat.stat-mechcs.CC

计算Sherrington-Kirkpatrick模型的配分函数在平均情况下是困难的

Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

David Gamarnik, Eren Kizildag

备注：: 31 pages
发表信息：: The Annals of Applied Probability 31(3): 1474-1504 (June 2021)
DOI：: 10.1214/20-AAP1625

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AI总结：

本文证明了自旋玻璃Sherrington-Kirkpatrick模型配分函数精确计算在平均情况下的计算复杂性，指出除非P=#P，否则不存在多项式时间算法，并在有限精度和实值计算模型下均确立了该问题的难度。

摘要：

我们确立了具有高斯耦合和随机外场的自旋玻璃Sherrington-Kirkpatrick模型相关配分函数精确计算的算法问题在平均情况下的计算复杂性。具体而言，我们证明了除非P=#P，否则不存在多项式时间算法能在平均情况下精确计算该配分函数。这是通过证明以下结论实现的：如果存在一个多项式时间算法，能对所有输入中多项式倒数比例（1/n^{O(1)}）的输入精确计算配分函数，那么就存在一个多项式时间算法，能以高概率对所有输入精确计算配分函数，从而得出P=#P。我们采用的计算模型是有限精度算术，其中算法输入首先被截断到数字精度为N的特定水平。我们证明的要素包括：带有随机外场的配分函数的随机和向下自可归约性；Cai等人用于建立矩阵积和式计算平均情况难度的论证；Sudan的列表解码算法，用于重构在足够多点上与给定数字列表相交的多项式；以及模大素数p的对数正态分布的近均匀性。据我们所知，我们的结果是首个确立自旋玻璃领域中出现的模型具有可证明难度的结果。此外，我们将结果扩展到了不同实值计算模型下的相同问题，例如使用在实值输入上运行的Blum-Shub-Smale机器。

英文摘要：

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless $P= \#P$, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction ($1/n^{O(1)}$) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $P=\#P$. The computational model that we adopt is {\em finite-precision arithmetic}, where the algorithmic inputs are truncated first to a certain level $N$ of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. \cite{cai1999hardness} for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan \cite{sudan1996maximum}, for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime $p$. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different {\em real-valued} computational model, e.g. using a Blum-Shub-Smale machine \cite{blum1988theory} operating over real-valued inputs.

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