Recently, Man\v{c}inska and Roberson proved that two graphs $G$ and $G'$ 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 $\mathcal{F}'$ of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case where each of $\mathcal{F}$ and $\mathcal{F}'$ 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 of constraint functions $\mathcal{F}$, and characterize the intertwiners of Qut$(\mathcal{F})$ as the signature matrices of planar Holant$(\mathcal{F}\,|\,\mathcal{EQ})$ quantum gadgets. Then we define a new notion of (projective) connectivity for constraint functions and reduce arity while preserving the quantum automorphism group. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lov\'asz in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.

翻译：最近，Mančinska 和 Roberson 证明了两个图 $G$ 与 $G'$ 量子同构当且仅当它们拥有来自所有平面图的同态计数相同。我们将此结果推广至平面 #CSP，其中 $\mathcal{F}$ 与 $\mathcal{F}'$ 为任意实值、任意元数约束函数集。图同态是特例，其中 $\mathcal{F}$ 与 $\mathcal{F}'$ 各仅含单个对称的 0-1 值二元约束函数。我们采用平面 Holant 问题的框架进行论证。为证明量子同构的约束函数集在任意平面 #CSP 实例上给出相同值，我们运用 Valiant 全息变换的一种新形式，该变换利用了定义量子同构的量子置换矩阵 $\mathcal{U}$。由于 $\mathcal{U}$ 元素的不交换性，此形式的全息变换仅适用于平面 Holant。为证明逆命题，我们引入约束函数集 $\mathcal{F}$ 的量子自同构群 Qut$(\mathcal{F})$，并将 Qut$(\mathcal{F})$ 的 intertwiner 刻画为平面 Holant$(\mathcal{F}\,|\,\mathcal{EQ})$ 量子器件的签名矩阵。随后，我们定义了约束函数的一种新（射影）连通性概念，并在保持量子自同构群不变的同时降低元数。最后，为应对从 0-1 值约束函数推广至实值约束函数带来的挑战，我们将 Lovász 在经典实权图同构中的一种技术改编至量子同构场景。