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

We establish the average-case hardness of algorithmic problem exact computation partition function associated with Sherrington–Kirkpatrick model spin glasses Gaussian couplings and random external field. In particular, we that unless P=#P, there does not exist a polynomial-time algorithm to exactly compute on average. This is done by showing if exists polynomial time algorithm, which computes for inverse fraction (1/nO(1)) all inputs, then high probability, yielding P=#P. The computational adopt finite-precision arithmetic, where inputs are truncated first certain level N digital precision. ingredients our proof include downward self-reducibility field; an argument Cai et al. (In STACS 99 (Trier) (1999) 90–99 Springer) establishing computing permanent matrix; list-decoding Sudan 37th Annual Symposium Foundations Computer Science (Burlington, VT, 1996) (1996) 164–172 IEEE Comput. Soc. Press), reconstructing polynomials intersecting given list numbers at sufficiently many points; near-uniformity log-normal distribution, modulo large prime p. To best knowledge, result one provable arising in field glasses. Furthermore, extend same under different real-valued model, example, using Blum–Shub–Smale machine [Proceedings 1988] 29th (1988) 387–397 IEEE) operating over inputs. that, 34+1nO(1) Our uses function, together control total variation distance variables presence convex perturbation, Berlekamp–Welch algorithm.