2 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
Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer
Two objects can be distinguished if they have different measurable
properties. Thus, distinguishability depends on the Physics of the objects. In
considering graphs, we revisit the Ising model as a framework to define
physically meaningful spectral invariants. In this context, we introduce a
family of refinements of the classical spectrum and consider the quantum
partition function. We demonstrate that the energy spectrum of the quantum
Ising Hamiltonian is a stronger invariant than the classical one without
refinements. For the purpose of implementing the related physical systems, we
perform experiments on a programmable annealer with superconducting flux
technology. Departing from the paradigm of adiabatic computation, we take
advantage of a noisy evolution of the device to generate statistics of low
energy states. The graphs considered in the experiments have the same classical
partition functions, but different quantum spectra. The data obtained from the
annealer distinguish non-isomorphic graphs via information contained in the
classical refinements of the functions but not via the differences in the
quantum spectra.Comment: 13 pages, 10 figure