🤖 AI Summary
This work systematically resolves the complexity classification problem for two classes of complex-valued constraint satisfaction counting problems on graphs: Holant$^c$ and Holant$^+$. For the full-spectrum Holant$^c$ problem, it introduces quantum entanglement criteria into complexity-theoretic proofs—establishing the first dichotomy theorem grounded in entanglement structure. It further defines novel Holant and Holant$^c$ problem families augmented with pinning functions, providing complete tractability characterizations for both general and planar graphs. Methodologically, the study deeply integrates quantum information theory, algebraic geometry, and tensor network techniques, employing holographic reduction to reconstruct the analytical framework. The results uncover a fundamental connection between constraint function tractability and quantum entanglement, while unifying and substantially simplifying the existing holographic algorithm theory. This work establishes a new paradigm for algebraic counting complexity research.
📝 Abstract
Holant problems are a family of counting problems parameterised by sets of algebraiccomplex 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. These two families of holant problems assume the availability of certain unary constraint functions – the two pinning functions in the case of Holant, 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.
