Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows to compare different entanglement structures and should lead to more efficient tensor network representations and contraction algorithms.

中文翻译：

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张量网络的资源理论


张量网络提供了量子多体态的简洁表示，是强相关量子系统的重要计算工具。它们的表达和计算能力的特点是底层纠缠结构，位于格子或更普遍的（超）图上，具有与（超）边缘关联的虚拟纠缠对或多部分纠缠状态。将这种底层纠缠结构更改为另一种结构可以带来理论和计算方面的好处。我们研究了一种自然资源理论，该理论使用多方纠缠将键维度的概念推广到纠缠结构。它是在多部分纠缠和代数复杂性理论的背景下研究的张量资源理论的直接扩展，允许将这些领域开发的复杂方法应用于张量网络。张量网络的资源理论既涉及量子多体态的局部纠缠结构，也涉及使用这种纠缠结构的张量网络收缩的（代数）复杂性。我们表明，纠缠结构之间存在超越逐边转换的转换，突出了我们资源理论的效率增益，这些增益反映了在寻找更好的矩阵乘法算法时获得的效率增益。我们还通过扩展最初在代数复杂性理论中开发的用于获得复杂性下限的各种方法来为这种转换的存在提供障碍。张量网络的资源理论允许比较不同的纠缠结构，并应该导致更有效的张量网络表示和收缩算法。