A Collapse Theorem for Holographic Algorithms with Matchgates on Domain Size at Most 4

TL;DR

This paper proves a sharp collapse theorem for holographic algorithms with matchgates, showing that for domain sizes 3 and 4, the basis size reduces to 1 and 2 respectively, simplifying their structure.

Contribution

It establishes a new basis collapse theorem for holographic algorithms with matchgates on domain sizes up to 4, revealing fundamental limitations and simplifications.

Findings

Basis size collapses to 1 for domain size 3

Basis size collapses to 2 for domain size 4

Provides new proof techniques using matchgates identities and group properties

Abstract

Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction . The two fundamental parameters of a holographic reduction are the domain size $k$ of the underlying problem, and the basis size $\ell$. A holographic reduction transforms the computation to matchgates by a linear transformation that maps to (a tensor product space of) a linear space of dimension $2^{\ell}$. We prove a sharp basis collapse theorem, that shows that for domain size 3 and 4, all non-trivial holographic reductions have basis size $\ell$ collapse to 1 and 2 respectively. The main proof techniques are Matchgates Identities, and a Group Property of matchgates signatures.