Holographaic Alogorithms on Bases of Rank 2

TL;DR

This paper extends the theory of holographic algorithms by reducing the basis transformation problem from higher domain sizes to size 2 for bases of rank 2, and generalizes the collapse theorem to larger domains.

Contribution

It introduces a reduction of the basis transformation problem from domain size k>2 to size 2 and generalizes the collapse theorem for bases of rank 2.

Findings

Reduced SRP on domain size k>2 to size 2

Generalized collapse theorem to domain size k>2

Established theoretical foundations for larger domain holographic algorithms

Abstract

An essential problem in the design of holographic algorithms is to decide whether the required signatures can be realized by matchgates under a suitable basis transformation (SRP). For holographic algorithms on domain size 2, [1, 2, 4, 5] have built a systematical theory. In this paper, we reduce SRP on domain size k>2 to SRP on domain size 2 for holographic algorithms on bases of rank 2. Furthermore, we generalize the collapse theorem of [3] to domain size k>2.