Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average
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1810.05907.pdf
Description
Submitted version
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374.95 KB
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Adobe PDF
Checksum (MD5)
0783f2cb2b366eaf40b58ab820b1de31
Author(s)
Gamarnik, David
Kizildag, Eren C
Date Issued
2020
Journal
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
Abstract
© 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.
MIT Department
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 Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/ISIT44484.2020.9174373
