A Dichotomy for Boolean Complex Holant Problems with Conjugate-Closed Signature Sets

2026-09-11

AI Summary

研究了闭合于共轭的布尔复值Holant问题，通过Xia的投影二元群框架和量子纠缠理论证明了复杂性二分法，扩展了实值Holant问题的结果。

Abstract

We study Boolean Holant problems with complex-valued signature sets closed under conjugation. Such sets arise naturally in tensor-network expressions for classical strong simulation of quantum circuits. We prove a complexity dichotomy for such problems with an explicit tractability criterion. This extends the dichotomy for real-valued Holant problems, with the same four tractability conditions. Our proofs use Xia's projective binary group framework and quantum entanglement theory. The conjugate closure assumption precisely makes $k$-uniformity, directly applicable to the classification of Holant problems, by realizing reduced density matrices via Holant gadgets. We also use the classification of absolutely maximally entangled states to resolve a particular $6$-ary obstruction in our inductive proof of the \#P-hardness.

Problem

• Boolean Holant problems
• conjugate-closed signature sets
• complexity dichotomy
• tensor-network expressions
• quantum circuits

Innovation

• complex-valued Holant problems
• conjugate-closed signature sets
• projective binary group framework
• quantum entanglement theory
• reduced density matrices