Generalized Counting Constraint Satisfaction Problems With Determinantal Circuits

TL;DR

This paper introduces determinantal circuits, a new class of computational circuits based on determinants, which expand the expressive power for counting problems and have applications in algebraic graph theory and quantum circuit simulation.

Contribution

It presents determinantal circuits with gates represented by minors of matrices, expanding the framework of Holant problems and matchgates, and shows they can be simulated efficiently by Pfaffian circuits.

Findings

Determinantal circuits can simulate Pfaffian circuits at quadratic cost.

Applications include proofs of algebraic graph theory theorems and quantum circuit simulation.

Provides a categorical framework for counting problems.

Abstract

Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomial-time algorithms for such problems include matchgates and matchcircuits, which are based on Pfaffians. In particular, they use gates which are expressible in terms of a vector of sub-Pfaffians of a skew-symmetric matrix. We introduce a new type of circuit based instead on determinants, with seemingly different expressive power. In these determinantal circuits, a gate is represented by the vector of all minors of an arbitrary matrix. Determinantal circuits permit a different class of gates. Applications of these circuits include proofs of theorems from algebraic graph theory including the Chung-Langlands formula for the number of rooted spanning forests of a graph and computing Tutte Polynomials of certain matroids. They also give a strategy for simulating quantum circuits with closed timelike curves. Monoidal category theory provides a useful language for discussing such counting problems, turning combinatorial restrictions into categorical properties. We introduce the counting problem in monoidal categories and count-preserving functors as a way to study FP subclasses of problems in settings which are generally #P-hard. Using this machinery we show that, surprisingly, determinantal circuits can be simulated by Pfaffian circuits at quadratic cost.