Holographic algorithms without matchgates

Abstract

The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.

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“…Substantial work has been built on top of MGI in the signature theory of matchgates [7,6,17,5,16,15,8,18,9,13]. In particular, a number of complexity dichotomy theorems have been proved that use this understanding of what matchgates can and cannot compute.…”

Tensors masquerading as matchgates: Relaxing planarity restrictions on Pfaffian circuits

“…As previously mentioned, for every n, P n and P _ n form varieties. We now briefly discuss the equations defining these varieties, as done in [10], based off of Valiants original identities [18]. Since the SWAP gate is in either pC 2 q b4 or ppC 2 q˚q b4 , we will thoroughly examine these particular cases.…”

“…One of the main motivations for matchcircuits is that they are akin to how quantum circuits are defined, but can be computed efficiently [8]. This relationship is especially clear when matchcircuits are place within the formalism of tensor networks (we shall assume the reader is familiar with these objects) [10,14].…”

Polynomial-time solvable #CSP problems via algebraic models and Pfaffian circuits

“…The Pfaffian circuit evaluation theorem relies on a valid edge order (Landsberg et al, 2013), here the planar spanning tree edge order.…”

“…Pfaffian circuits (Landsberg et al, 2013;Morton, 2011;Morton and Turner, 2015) are a simplified and extensible reformulation of Valiant's notion of a holographic algorithm, which builds on J.Y. Cai and V. Choudhary's work in expressing holographic algorithms in terms of tensor contraction networks (Cai and Choudhary, 2007).…”