Complex-valued Holant with a non-trivial odd-arity signature is classified: every instance is either #P-hard or in FPNP.
From Holant to Quantum Entanglement and Back
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory. We discover two particular entangled states $|{\Psi_6}\rangle$ of 6 qubits and $|{\Psi_8}\rangle$ of 8 qubits respectively, that have extraordinary and unique closure properties in terms of the Bell property. Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing an odd-arity signature. The signatures need not be symmetric, and no auxiliary signatures are assumed.
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From an odd arity signature to a Holant dichotomy
Complex-valued Holant with a non-trivial odd-arity signature is classified: every instance is either #P-hard or in FPNP.