A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

Holant problems are a family of counting problems parameterized 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.