5 research outputs found
Computing the partition function of the Sherrington-Kirkpatrick model is hard on average
We establish the average-case hardness of the algorithmic problem of exact
computation of the partition function associated with the
Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and
random external field. In particular, we establish that unless , there
does not exist a polynomial-time algorithm to exactly compute the partition
function on average. This is done by showing that if there exists a polynomial
time algorithm, which exactly computes the partition function for inverse
polynomial fraction () of all inputs, then there is a polynomial
time algorithm, which exactly computes the partition function for all inputs,
with high probability, yielding . The computational model that we adopt
is {\em finite-precision arithmetic}, where the algorithmic inputs are
truncated first to a certain level of digital precision. The ingredients of
our proof include the random and downward self-reducibility of the partition
function with random external field; an argument of Cai et al.
\cite{cai1999hardness} for establishing the average-case hardness of computing
the permanent of a matrix; a list-decoding algorithm of Sudan
\cite{sudan1996maximum}, for reconstructing polynomials intersecting a given
list of numbers at sufficiently many points; and near-uniformity of the
log-normal distribution, modulo a large prime . To the best of our
knowledge, our result is the first one establishing a provable hardness of a
model arising in the field of spin glasses.
Furthermore, we extend our result to the same problem under a different {\em
real-valued} computational model, e.g. using a Blum-Shub-Smale machine
\cite{blum1988theory} operating over real-valued inputs.Comment: 31 page
Computing the partition function of the Sherrington–Kirkpatrick model is hard on average
© 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
Computing the Partition Function of the Sherrington-Kirkpatrick Model is Hard on Average
© 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