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$G$和$G'$是量子同构的当且仅当它们从所有平面图中接受相同数量的同态。我们将这一结果推广到平面#CSP，其中包含任意一对实数值的任意元约束函数集$\mathcal{F}$和$\mathcal{F}'$。图同态是每个$\mathcal{F}$和$\mathcal{F}'$仅包含一个对称的0-1值二元约束函数的特例。我们的研究采用平面Holant问题的框架。为了证明量子同构的约束函数集在任何平面#CSP实例上给出相同的值，我们应用了Valiant的全息变换的一种新形式，使用定义量子同构的量子置换矩阵$\mathcal{U}$。由于$\mathcal{U}$的条目的非交换性，这种全息变换形式仅适用于平面Holant。为了证明逆命题，我们引入了约束函数集$\mathcal{F}$的量子自同构群Qut$(\mathcal{F})$，并将Qut$(\mathcal{F})$的交错子刻画为平面Holant$(\mathcal{F}\,|\,\mathcal{EQ})$量子小工具的签名矩阵。然后，我们为约束函数定义了新的（射影）连通性概念，并在保持量子自同构群的同时降低元数。最后，为了应对从0-1值约束函数推广到实数值约束函数所带来的挑战，我们将Lovász在经典背景下用于实加权图同构的技术调整到量子同构的背景下。