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Partition functions of certain classes of "spin glass" models in statisticalphysics show strong connections to combinatorial graph invariants. Also knownas homomorphism functions they allow for the representation of many suchinvariants, for example, the number of independent sets of a graph or thenumber nowhere zero k-flows. Contributing to recent developments on thecomplexity of partition functions we study the complexity of partitionfunctions with complex values. These functions are usually determined by asquare matrix A and it was shown by Goldberg, Grohe, Jerrum, and Thurley thatfor each real-valued symmetric matrix, the corresponding partition function iseither polynomial time computable or #P-hard. Extending this result, we give acomplete description of the complexity of partition functions definable byHermitian matrices. These can also be classified into polynomial timecomputable and #P-hard ones. Although the criterion for polynomial timecomputability is not describable in a single line, we give a clear account ofit in terms of structures associated with Abelian groups.

The Complexity of Partition Functions on Hermitian Matrices