Some Results on Matchgates and Holographic Algorithms

Abstract

We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [5], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plücker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

On Symmetric Signatures in Holographic Algorithms

“…We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results [4] where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. This in particular confirms a conjecture by Valiant [18].…”

“…Our characterization is valid for all fields with sufficiently large characteristic p (including the complex numbers C). These improve previous results [4] where only symmetric signatures over the Hadamard basis, which is a special basis of size 1, were obtained. In [4], those results were proved using properties of Krawtchouk polynomials.…”

“…These improve previous results [4] where only symmetric signatures over the Hadamard basis, which is a special basis of size 1, were obtained. In [4], those results were proved using properties of Krawtchouk polynomials. Here we are able to prove a much stronger results without the use of these special polynomials.…”

Valiant’s Holant Theorem and Matchgate Tensors

“…Also he proved an interesting lower bound. This lower bound proof also uses the realizability results from [2,1]. In [3], we showed that this particular restrictive SAT counting problem can also be solved by a holographic algorithm using a basis of size 1.…”

“…Before this paper Valiant already gave matchgate identities [15]. In [1] we achieved a unification of the planarmatchgate/signature theory and the generals-matchgate/character theory, and together with [2] this gives a complete algebraic characterization of all realizable signatures. The general-matchgate/character theory was proposed by Valiant in [16].…”