A full dichotomy for Holant$^c$, inspired by quantum computation

TL;DR

This paper uses quantum information theory to classify the computational complexity of Holant problems, introducing new dichotomies for Holant$^+$ and Holant$^c$, and connecting quantum states with problem complexity.

Contribution

It provides the first full complexity classification (dichotomy) for Holant$^c$, based on quantum information theory, and introduces a new family Holant$^+$ with its own dichotomy.

Findings

Derived a full dichotomy for Holant$^c$ problems.

Established a dichotomy for Holant$^+$, including planar cases.

Proved an original result about entangled quantum states.

Abstract

Holant problems are a family of counting problems parameterised by sets of algebraic-complex valued constraint functions, and defined on graphs. They arise from the theory of holographic algorithms, which was originally inspired by concepts from quantum computation. Here, we employ quantum information theory to explain existing results about holant problems in a concise way and to derive two new dichotomies: one for a new family of problems, which we call Holant$^+$, and, building on this, a full dichotomy for Holant$^c$. These two families of holant problems assume the availability of certain unary constraint functions -- the two pinning functions in the case of Holant$^c$, and four functions in the case of Holant$^+$ -- and allow arbitrary sets of algebraic-complex valued constraint functions otherwise. The dichotomy for Holant$^+$ also applies when inputs are restricted to instances defined on planar graphs. In proving these complexity classifications, we derive an original result about entangled quantum states.