The reconstruction of quantum states from experimental measurements, often achieved using quantum state tomography (QST), is crucial for the verification and benchmarking of quantum devices. However, performing QST for a generic unstructured quantum state requires an enormous number of state copies that grows \emph{exponentially} with the number of individual quanta in the system, even for the most optimal measurement settings. Fortunately, many physical quantum states, such as states generated by noisy, intermediate-scale quantum computers, are usually structured. In one dimension, such states are expected to be well approximated by matrix product operators (MPOs) with a finite matrix/bond dimension independent of the number of qubits, therefore enabling efficient state representation. Nevertheless, it is still unclear whether efficient QST can be performed for these states in general. In this paper, we attempt to bridge this gap and establish theoretical guarantees for the stable recovery of MPOs using tools from compressive sensing and the theory of empirical processes. We begin by studying two types of random measurement settings: Gaussian measurements and Haar random rank-one Positive Operator Valued Measures (POVMs). We show that the information contained in an MPO with a finite bond dimension can be preserved using a number of random measurements that depends only \emph{linearly} on the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with physical quantum measurements through Haar random rank-one POVMs that can be implemented on quantum computers. We prove that only a \emph{polynomial} number of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state.

翻译：从实验测量中重建量子态（通常采用量子态层析技术）是量子器件验证与基准测试的关键环节。然而，对一般非结构化量子态进行量子态层析，即便采用最优测量设置，也需要随系统量子比特数呈指数增长的大量态副本。幸运的是，许多物理量子态（如有噪中等规模量子计算机生成的量子态）通常具有结构化特征。在一维情况下，此类态可由有限矩阵/键维数（与量子比特数无关）的矩阵乘积算符（MPO）良好近似，从而实现高效态表示。但此类量子态能否普遍实现高效量子态层析仍不明确。本文尝试弥合这一理论空白，利用压缩感知与经验过程理论为MPO的稳定恢复建立理论保证。我们首先研究两类随机测量设置：高斯测量与Haar随机秩一正算子值测度（POVM）。研究表明，在无测量统计误差假设下，可仅使用与量子比特数呈线性关系的随机测量次数来保持有限键维数MPO所包含的信息。继而研究基于Haar随机秩一POVM（可在量子计算机上实现）的MPO量子态层析方法。我们证明，仅需随量子比特数呈多项式增长的态副本数即可保证MPO态恢复误差的有界性。