3,510 research outputs found
Complexity of Ising Polynomials
This paper deals with the partition function of the Ising model from
statistical mechanics, which is used to study phase transitions in physical
systems. A special case of interest is that of the Ising model with constant
energies and external field. One may consider such an Ising system as a simple
graph together with vertex and edge weights. When these weights are considered
indeterminates, the partition function for the constant case is a trivariate
polynomial Z(G;x,y,z). This polynomial was studied with respect to its
approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003.
Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied by D.
Andr\'{e}n and K. Markstr\"{o}m in 2009.
We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that
of the Tutte polynomial, which is well-known to be closely related to the Potts
model in the absence of an external field. We show that Z(G;\x,\y,\z) is
#P-hard to evaluate at all points in , except those in an
exception set of low dimension, even when restricted to simple graphs which are
bipartite and planar. A counting version of the Exponential Time Hypothesis,
#ETH, was introduced by H. Dell, T. Husfeldt and M. Wahl\'{e}n in 2010 in order
to study the complexity of the Tutte polynomial. In analogy to their results,
we give a dichotomy theorem stating that evaluations of Z(G;t,y) either take
exponential time in the number of vertices of to compute, or can be done in
polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in
polynomial time on graphs of bounded clique-width, which is not known in the
case of the Tutte polynomial
Learning Graphical Models Using Multiplicative Weights
We give a simple, multiplicative-weight update algorithm for learning
undirected graphical models or Markov random fields (MRFs). The approach is
new, and for the well-studied case of Ising models or Boltzmann machines, we
obtain an algorithm that uses a nearly optimal number of samples and has
quadratic running time (up to logarithmic factors), subsuming and improving on
all prior work. Additionally, we give the first efficient algorithm for
learning Ising models over general alphabets.
Our main application is an algorithm for learning the structure of t-wise
MRFs with nearly-optimal sample complexity (up to polynomial losses in
necessary terms that depend on the weights) and running time that is
. In addition, given samples, we can also learn the
parameters of the model and generate a hypothesis that is close in statistical
distance to the true MRF. All prior work runs in time for
graphs of bounded degree d and does not generate a hypothesis close in
statistical distance even for t=3. We observe that our runtime has the correct
dependence on n and t assuming the hardness of learning sparse parities with
noise.
Our algorithm--the Sparsitron-- is easy to implement (has only one parameter)
and holds in the on-line setting. Its analysis applies a regret bound from
Freund and Schapire's classic Hedge algorithm. It also gives the first solution
to the problem of learning sparse Generalized Linear Models (GLMs)
On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants
It is shown that the canonical problem of classical statistical
thermodynamics, the computation of the partition function, is in the case of
+/-J Ising spin glasses a particular instance of certain simple sums known as
quadratically signed weight enumerators (QWGTs). On the other hand it is known
that quantum computing is polynomially equivalent to classical probabilistic
computing with an oracle for estimating QWGTs. This suggests a connection
between the partition function estimation problem for spin glasses and quantum
computation. This connection extends to knots and graph theory via the
equivalence of the Kauffman polynomial and the partition function for the Potts
model.Comment: 8 pages, incl. 2 figures. v2: Substantially rewritte
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
We examine two proposals for marginally self-correcting quantum memory, the
cubic code by Haah and the welded code by Michnicki. In particular, we prove
explicitly that they are absent of topological order above zero temperature, as
their Gibbs ensembles can be prepared via a short-depth quantum circuit from
classical ensembles. Our proof technique naturally gives rise to the notion of
free energy associated with excitations. Further, we develop a framework for an
ergodic decomposition of Davies generators in CSS codes which enables formal
reduction to simpler classical memory problems. We then show that memory time
in the welded code is doubly exponential in inverse temperature via the Peierls
argument. These results introduce further connections between thermal
topological order and self-correction from the viewpoint of free energy and
quantum circuit depth.Comment: 19 pages, 18 figure
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