发表机构
University of Warwick; University of Oxford（华威大学; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究利用参数化图论，明确割宽等图参数对张量网络态的矩阵乘积态、树张量网络表示的约束，推导可实现及不可知张量网络层析的样本与计算复杂度上界，扩展了解缠学习器的适用范围。
AI 中文摘要
参数化图论研究图论问题的复杂度如何依赖于输入图的结构参数，这一视角已被证明可用于分析张量网络模拟（Markov和Shi，2008）。但其对张量网络表示与层析的影响尚未得到充分理解。具体而言，哪些图参数决定了张量网络态（TNS）是否存在可处理的矩阵乘积态（MPS）或树张量网络（TTN）表示，又有哪些参数控制该态的学习复杂度？我们利用参数化图论解决这些问题。首先，我们证明割宽和树割宽为将TNS表示为MPS或TTN所需的键维度开销提供了上界；在TTN情形下，树割宽还对分组子系统的局域维度构成约束，证明基于纠缠重路由（经典网络中信息重路由的张量网络类似物）。其次，我们推导了可实现TNS层析的样本复杂度与计算复杂度的图相关上界，其指数依赖于割宽、树割宽以及一个新图参数——学习复杂度（我们根据度和树宽对该参数进行了界定）；这些结果通过将Cramer等人（2010）提出的解缠MPS学习器（经Bakshi等人（2025）、Lin等人（2025）进一步分析）扩展至TTN及任意已知图上的张量网络而获得。最后，我们将该框架扩展至不可知情形：对于任意输入态，我们的不可知学习器输出一个纯态，其保真度与给定键维度的图上张量网络态的最优值相差加性误差ε，且样本复杂度与计算复杂度具有明确的图相关上界。
英文摘要
Parameterised graph theory studies how the complexity of graph-theoretic problems depends on structural parameters of the input graph. This perspective has proved useful in analysing tensor-network simulation (Markov and Shi, 2008). Its implications for tensor-network representations and tomography are less well understood. In particular, which graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state? We address these questions using parameterised graph theory. First, we show that cutwidth and tree-cutwidth bound the bond dimension overhead required to represent a TNS as an MPS or TTN. In the TTN case, tree-cutwidth also bounds the local dimension of the grouped subsystems. The proofs are based on entanglement rerouting, a tensor-network analogue of rerouting information in a classical network. Second, we derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth. We obtain these results by extending the disentangling MPS learner of (Cramer et al., 2010), as analysed further in (Bakshi et al., 2025; Lin et al., 2025), to TTNs and to tensor networks on arbitrary known graphs. Finally, we extend the framework beyond the realisable setting. For an arbitrary input state, our agnostic learner outputs a pure state whose fidelity is within additive error $ε$ of the optimum over tensor-network states on the given graph with a given bond dimension, with explicit graph-dependent bounds on sample and computational complexity.
Comments74 + 12 pages; 6 figures
