# Counting perfect matchings and the eight-vertex model

## TL;DR

This paper investigates the computational complexity of approximating the partition function of the eight-vertex model on 4-regular graphs, establishing connections to the hard problem of counting perfect matchings and providing new characterizations of matchgates.

## Contribution

It relates the approximability of the eight-vertex model to counting perfect matchings and characterizes nonnegative 4-ary matchgates, extending previous complexity results.

## Key findings

- Approximation of the partition function is as hard as counting perfect matchings in certain parameter regions.
- Computing the partition function can be reduced to counting perfect matchings in larger parameter regions.
- Identifies a parameter region where planar graph approximation is feasible but general graph approximation is as hard as counting perfect matchings.

## Abstract

We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in arXiv:1811.03126 [cs.CC]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the previously known FPRASable region, we show that computing the partition function can be reduced to (with or without approximation) counting perfect matchings. Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma. We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems of this kind.