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（通常通过量子态层析成像实现）对于量子器件的验证和基准测试至关重要。然而，对通用非结构化量子态进行量子态层析成像需要大量状态副本，该数量随系统中独立量子数量呈指数增长，即使采用最优测量设置也是如此。幸运的是，许多物理量子态（例如噪声中等规模量子计算机生成的态）通常具有结构性。在一维系统中，这类态可被矩阵乘积算符以有限矩阵/键维数（与量子比特数无关）良好近似，从而实现高效状态表示。尽管如此，这些态在一般情况下能否实现高效量子态层析成像仍不明确。本文致力于弥合这一差距，并利用压缩感知与经验过程理论为矩阵乘积算符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：高斯测量与哈达玛随机秩一正算子值测量。研究表明，在无测量统计误差的假设下，具有有限键维数的矩阵乘积算符所包含的信息可通过仅与量子比特数呈线性相关的随机测量次数得以保存。随后，我们进一步研究基于矩阵乘积算符的量子态层析成像，采用可在量子计算机上实现的哈达玛随机秩一正算子值测量进行物理量子测量。我们证明，仅需与量子比特数呈多项式关系的状态副本数量，即可保证矩阵乘积算符态在恢复误差有界。