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

Abstract

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 $\mathit{P}=\mathrm{\#}\mathit{P}$, 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 ($1/{\mathit{n}}^{\mathit{O}(1)}$) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $\mathit{P}=\mathrm{\#}\mathit{P}$. The computational model that we adopt is 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. (In STACS 99 (Trier) (1999) 90–99 Springer) for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan (In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) (1996) 164–172 IEEE Comput. Soc. Press), 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 real-valued computational model, for example, using a Blum–Shub–Smale machine (In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science (1988) 387–397 IEEE) operating over real-valued inputs. We establish that, if there exists a polynomial time algorithm which exactly computes the partition function for $\frac{3}{4}+\frac{1}{{\mathit{n}}^{\mathit{O}(1)}}$ fraction of all inputs, then there exists a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $\mathit{P}=\mathrm{\#}\mathit{P}$. Our proof uses the random self-reducibility of the partition function, together with a control over the total variation distance for log-normal random variables in presence of a convex perturbation, and the Berlekamp–Welch algorithm.