Contraction of matchgate tensor networks on non-planar graphs

TL;DR

This paper introduces a new method for contracting matchgate tensor networks on non-planar graphs using Grassmann variables, extending previous planar graph techniques to graphs of higher genus.

Contribution

It presents an alternative contraction approach for matchgate tensor networks that applies to non-planar graphs, utilizing Grassmann variables and Gaussian integrals.

Findings

Contraction time is polynomial in number of vertices and exponential in genus.

Method extends matchgate tensor network contraction to non-planar graphs.

Provides a computational complexity bound involving graph genus and edges to remove.

Abstract

A tensor network is a product of tensors associated with vertices of some graph $G$ such that every edge of $G$ represents a summation (contraction) over a matching pair of indexes. It was shown recently by Valiant, Cai, and Choudhary that tensor networks can be efficiently contracted on planar graphs if components of every tensor obey a system of quadratic equations known as matchgate identities. Such tensors are referred to as matchgate tensors. The present paper provides an alternative approach to contraction of matchgate tensor networks that easily extends to non-planar graphs. Specifically, it is shown that a matchgate tensor network on a graph $G$ of genus $g$ with $n$ vertices can be contracted in time $T=poly(n) + 2^{2g} O(m^3)$ where $m$ is the minimum number of edges one has to remove from $G$ in order to make it planar. Our approach makes use of anticommuting (Grassmann) variables and Gaussian integrals.