Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient framework for tensor representation by leveraging decision diagram techniques. However, the current implementation of TDDs and other decision diagrams fail to exploit tensor isomorphisms, limiting their compression potential. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDDs that incorporates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which uses Pauli operators for quantum states, LimTDD employs the $XP$-stabilizer group, enabling broader applicability across tensor-based tasks. We present efficient algorithms for normalization, slicing, addition, and contraction, critical for tensor network applications. Theoretical analysis demonstrates that LimTDDs achieve greater compactness than TDDs and, in best-case scenarios and for quantum state representations, offer exponential compression advantages over both TDDs and LIMDDs. Experimental results in quantum circuit tensor computation and simulation confirm LimTDD's superior efficiency. Open-source code is available at https://github.com/Veriqc/LimTDD.

翻译：张量网络作为高效表示和处理高维数据的强大工具，在量子物理、机器学习和数据压缩等领域具有广泛应用。张量决策图通过利用决策图技术，为张量表示提供了一个高效框架。然而，当前TDD及其他决策图的实现未能充分利用张量同构性，限制了其压缩潜力。本文提出了局部可逆映射张量决策图，作为TDD的扩展，通过引入局部可逆映射来实现更紧凑的表示。与使用Pauli算子表示量子态的LIMDD不同，LimTDD采用$XP$-稳定子群，从而在基于张量的任务中具有更广泛的适用性。我们提出了用于归一化、切片、加法及缩并的高效算法，这些算法对张量网络应用至关重要。理论分析表明，LimTDD相比TDD实现了更高的紧凑性，且在最佳情况下及量子态表示中，相比TDD和LIMDD均具有指数级压缩优势。在量子电路张量计算与仿真中的实验结果证实了LimTDD的卓越效率。开源代码发布于https://github.com/Veriqc/LimTDD。