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arXiv · 2609.13132

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

Jincheng Guan·Shuai Shao·Zhuxiao Tang
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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.

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Jincheng Guan, Shuai Shao, Zhuxiao Tang. 2026-09-11. A Dichotomy for Boolean Complex Holant Problems with Conjugate-Closed Signature Sets. https://arxiv.org/abs/2609.13132

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