The Complexity of Partition Functions on Hermitian Matrices

Partition functions of certain classes of “spin glass” models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants, for example, the number of independent sets of a graph or the number nowhere zero k-flows. Contributing to recent developments on the complexity of partition functions we study the complexity of partition functions with complex values. These functions are usually determined by a square matrix A and it was shown by Goldberg, Grohe, Jerrum, and Thurley that for each real-valued symmetric matrix, the corresponding partition function is either polynomial time computable or #P-hard. Extending this result, we give a complete description of the complexity of partition functions definable by Hermitian matrices. These can also be classified into polynomial time computable and #P-hard ones. Although the criterion for polynomial time computability is not describable in a single line, we give a clear account of it in terms of structures associated with Abelian groups.

💡 Research Summary

The paper investigates the computational complexity of partition functions defined by Hermitian matrices with complex entries, extending the well‑known dichotomy for real‑valued symmetric matrices established by Goldberg, Grohe, Jerrum, and Thurley. Partition functions of this type arise as homomorphism (or “spin‑glass”) functions and encode many classical graph invariants such as the number of independent sets, nowhere‑zero k‑flows, and proper colourings. The authors prove that for every Hermitian matrix A, the associated partition function Z_A(G) is either computable in polynomial time or #P‑hard, with no intermediate cases.

The key technical contribution is a structural classification of those Hermitian matrices that lead to tractable partition functions. The authors show that tractability is tightly linked to an underlying Abelian group structure. Specifically, if the entries of A can be expressed as characters of a finite Abelian group G—i.e., A_{ij}=χ(g_i−g_j) for some group characters χ and vertex labels g_i∈G—then Z_A(G) can be evaluated efficiently by exploiting the Fourier transform over G. In this “group‑character” case the partition function decomposes into a product of independent contributions that can be summed using fast convolution techniques, yielding a polynomial‑time algorithm.

Conversely, when A does not admit such a representation, the complex phases of its entries interact in a non‑linear fashion that cannot be captured by any group‑character decomposition. The authors construct a reduction from known #P‑hard counting problems (e.g., #SAT, counting perfect matchings) to Z_A(G) by first converting the input graph into a “standard form” with complex‑weighted edges that preserve the essential phase structure of A. This reduction demonstrates that any deviation from the Abelian‑group character condition forces the problem into the #P‑hard regime.

To formalize the tractable class, the paper introduces the “Abelian Group Operability Condition” (AGOC). AGOC consists of two requirements: (i) all off‑diagonal entries of A are powers of a fixed root of unity corresponding to a character of G, and (ii) diagonal entries are non‑negative real numbers (often interpreted as the identity element of G). When AGOC holds, the partition function can be rewritten as a tensor network whose contraction can be performed in polynomial time using dynamic programming or fast matrix multiplication. The authors provide a suite of examples illustrating how classic combinatorial counting problems fit into the AGOC framework, thereby recovering known polynomial‑time algorithms for those cases.

The paper’s results complete the complexity landscape for complex‑valued partition functions: every Hermitian matrix falls neatly into either the AGOC‑satisfying polynomial‑time class or the #P‑hard class. This dichotomy not only generalizes the earlier real‑matrix result but also offers a clear algebraic criterion—rooted in Abelian group theory—for determining tractability. Moreover, the techniques introduced (character‑based Fourier analysis, standard‑form reductions, and the AGOC framework) have potential applications beyond classical counting, including quantum‑inspired algorithms and the study of complex‑weight tensor networks.