Learning the closest matrix product state (MPS) representation of a quantum state enables useful tools for quantum machine learning and analysis of complex quantum systems. In this work, we study the problem of learning MPS in the following setting: given many copies of an input MPS, the task is to recover a classical description of the state. The best known polynomial-time algorithm, introduced by [LCLP10, CPF+10], requires linear circuit depth and $\widetilde O(n^5)$ samples, and has seen no improvement in over a decade. These costs, neither known to be optimal, renders existing algorithms impractical for near-term quantum devices with limited resources. We introduce parallel disentangling algorithms for MPS learning. For exact MPS learning, our algorithm runs in polynomial time and uses circuit depth $O(\log n)$ and sample complexity $\widetilde O(n^3)$, improving both the depth and the dependence on the system size $n$. The key idea is to exploit the bounded-rank structure of reduced states on middle blocks of an MPS and organize the disentangling operations in a tree structure. We further extend the algorithm to closest MPS learning, improving the sample complexity dependence on $n$ from $n^9$ to $n^7$ and complement the algorithms with an $Ω(n)$ product-state lower bound. We also investigate MPS learning under hardware constraints, including restricted measurements and geometric connectivity. Under the Learning Parity with Noise (LPN) assumption, we show computational hardness for learning an MPS(2) family with non-adaptive single-qubit measurements. Finally, we show that our algorithm can be implemented with depth $O(q n^{1/q})$ on a $q$-dimensional hypercubic lattice, giving an asymptotic reduction in depth. Together, our work provides a complete characterization of the quantum resources needed for efficient MPS learning.

翻译：学习量子态最接近的矩阵乘积态（MPS）表示，为量子机器学习及复杂量子系统分析提供了实用工具。本文研究如下设定中的MPS学习问题：给定输入MPS的多个副本，任务是恢复该状态的经典描述。目前已知最佳多项式时间算法由[LCLP10, CPF+10]提出，需要线性电路深度和$\widetilde O(n^5)$个样本，且十余年来未获改进。这些代价未知是否为最优，使得现有算法难以应用于资源受限的近量子器件。我们引入MPS学习的并行解纠缠算法。对于精确MPS学习，算法运行于多项式时间，使用电路深度$O(\log n)$和样本复杂度$\widetilde O(n^3)$，在深度和系统规模$n$的依赖关系上均实现优化。核心思想是利用MPS中间分块约化态的有界秩结构，并将解纠缠操作组织成树形结构。进一步，我们将算法扩展到最近MPS学习，将样本复杂度对$n$的依赖关系从$n^9$降至$n^7$，并辅以$\Omega(n)$乘积态下界。我们还研究了硬件约束下的MPS学习，包括受限测量和几何连接性。在带噪声学习悖论（LPN）假设下，我们证明了对MPS(2)族使用非自适应单比特测量进行学习具有计算困难性。最后，我们证明算法可在$q$维超立方体晶格上用深度$O(q n^{1/q})$实现，深度渐近降低。综上，本研究为高效MPS学习所需的量子资源提供了完整刻画。