|Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average
|David Gamarnik, Eren C. Kizildag, MIT, United States
|T.1: Complexity and Computation Theory
|Topics in Information Theory
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|We establish the average-case hardness of the algorithmic problem of exactly computing the partition function of the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings. In particular, we establish that unless $P= \#P$, there does not exist a polynomial-time algorithm to exactly compute this object on average. This is done by showing that if there exists a polynomial-time algorithm exactly computing the partition function for a certain fraction of all inputs, then there is a polynomial-time algorithm exactly computing this object for all inputs, with high probability, yielding $P=\#P$. Our results cover both finite-precision arithmetic as well as the real-valued computational models. The ingredients of our proofs include Berlekamp-Welch algorithm, a list-decoding algorithm by Sudan for reconstructing a polynomial from its noisy samples, near-uniformity of log-normal distribution modulo a large prime; and a control over total variation distance for log-normal distribution under convex perturbation. To the best of our knowledge, this is the first average-case hardness result pertaining a statistical physics model with random parameters.