A complete dichotomy for complex-valued Holant^c

Holant problems are a family of counting problems on graphs, parametrised by sets of complexvalued functions of Boolean inputs. HOLANTc 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 HOLANTc and one for HOLANTc with complex symmetric functions. Here, we derive a dichotomy theorem for HOLANTc with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued HOLANTc dichotomy. The proof uses results from quantum information theory, particularly about entanglement.