A new Holant dichotomy inspired by quantum computation

TL;DR

This paper introduces a new Holant problem family called Holant$^+$, using quantum information theory to establish a complete complexity dichotomy, bridging known Holant classes and advancing understanding of counting problems.

Contribution

It derives a full dichotomy theorem for Holant$^+$ problems by applying quantum entanglement theory, expanding the scope of Holant complexity classifications.

Findings

Established a full complexity dichotomy for Holant$^+$.

Connected quantum entanglement concepts with Holant problem analysis.

First Holant dichotomy with unrestricted functions and finite freely available functions.

Abstract

Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many other counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problem instances into those which can be solved in polynomial time and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before. Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant$^+$. This family sits in between the known families of Holant$^*$, for which a full dichotomy is known, and Holant$^c$, for which only a restricted dichotomy is known. Using knowledge from entanglement theory -- both previously existing work and new results of our own -- we prove a full dichotomy theorem for Holant$^+$, which is very similar to the restricted Holant$^c$ dichotomy. Other than the dichotomy for #R$_3$-CSP, ours is the first Holant dichotomy in which the allowed functions are not restricted and in which only a finite number of functions are assumed to be freely available.