Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average
Author(s)
Gamarnik, David; Kizildag, Eren C
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© 2020 IEEE. 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.
Date issued
2020Department
Sloan School of Management; Massachusetts Institute of Technology. Institute for Data, Systems, and Society; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer ScienceJournal
IEEE International Symposium on Information Theory - Proceedings
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Gamarnik, David and Kizildag, Eren C. 2020. "Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average." IEEE International Symposium on Information Theory - Proceedings, 2020-June.
Version: Original manuscript