# A complete dichotomy for complex-valued Holant^c

## TL;DR

This paper establishes a complete complexity classification for Holant^c problems with complex-valued functions, extending previous results to non-symmetric functions using quantum information theory techniques.

## Contribution

It provides the first dichotomy theorem for complex-valued, non-symmetric Holant^c problems, broadening the understanding of their computational complexity.

## Key findings

- Identifies tractable cases as complex generalisations of real-valued Holant^c.
- Proves #P-hardness for all other cases.
- Uses quantum entanglement concepts in the proof.

## Abstract

Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement.

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Source: https://tomesphere.com/paper/1704.05798