We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

中文翻译：

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带域墙的字符串网络中的矩阵乘积运算符对称性和交织器

我们以字符串网络模型的投影纠缠对状态（PEPS）表示形式提供了虚拟非局部矩阵乘积运算符（MPO）对称性的描述。给定这样的PEPS表示，我们证明了其MPO对称性的一致性条件总计可以由双模块类别的五边形方程识别的六个耦合方程组。这使我们能够对所有等效的PEPS表示进行分类，并在它们之间建立MPO交织器，从而综合和概括拓扑阶段的多种张量网络表示。此外，我们使用这种概括来构建由Kitaev和Kong [Commun。数学。物理 313（2012）351-373]。尽管流行的抽象分类方法足以描述拓扑阶段的结构，但需要显式张量网络表示形式才能在计算机上模拟这些系统，例如，基于带边界的字符串网络计算量子纠错码的阈值时，就需要使用显式张量网络表示。最后，我们证明所有这些字符串网络PEPS表示形式都可以理解为具有物理边界的三流形上拓扑场论的Turaev-Viro状态和模型的特定实例，从而将这些张量网络构造置于数学上严格的基础上。例如基于带边界的字符串网络计算量子纠错码的阈值所需要的。最后，我们证明所有这些字符串网络PEPS表示形式都可以理解为具有物理边界的三流形上拓扑场论的Turaev-Viro状态和模型的特定实例，从而将这些张量网络构造置于数学上严格的基础上。例如基于带边界的字符串网络计算量子纠错码的阈值所需要的。最后，我们证明所有这些字符串网络PEPS表示形式都可以理解为具有物理边界的三流形上拓扑场论的Turaev-Viro状态和模型的特定实例，从而将这些张量网络构造置于数学上严格的基础。