Planar #CSP Equality Corresponds to Quantum Isomorphism – A Holant Viewpoint

Recently, Mančinska and Roberson proved that two graphs G and \(G^{\prime }\) are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix \(\mathcal {U}\) defining the quantum isomorphism. Due to the noncommutativity of \(\mathcal {U}\) ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group \(Qut(\mathcal {F})\) of a set \(\mathcal {F}\) of constraint functions/tensors and characterize the intertwiners of \(Qut(\mathcal {F})\) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.