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 [Lip91] 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.
The random self-reducibility of polynomials was also used by Gamarnik REF in order to prove that exactly computing the partition function of the Sherrington-Kirkpatrick model in statistical physics is hard on average.
Computing the partition function of the Sherrington-Kirkpatrick model is hard on average
ArXiv
ArXiv
Mathematics Physics Computer Science