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

David Gamarnik

Published 2018-10-13Version 1

We consider the algorithmic problem of computing the partition function of the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings. We show that there is no polynomial time algorithm for computing the partition function exactly (in the sense to be made precise), unless P=\#P. Our proof uses the Lipton's reducibility trick of computation modulo large primes~\cite{lipton1991new} and near-uniformity of the log-normal distribution in small intervals. To the best of our knowledge, this is the first statistical physics model with random parameters for which such average case hardness is established.