We introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.

翻译：本文提出草图层析成像，一种基于经典阴影协议用于量子可观测量估计的高效量子态层析方法。该方法适用于真实量子态为矩阵乘积态的情况。在MPS假设下，真实态的密度矩阵可采用张量列车拟设表示，我们通过一系列可观测量估计来拟设中的张量分量进行估计，从而输出密度矩阵的近似。该方法被证明具有收敛性，且样本复杂度随系统规模呈二次方增长。我们进行了大量数值实验，表明该方法能输出量子态的高精度近似。对于涉及中等规模子系统的可观测量估计任务，我们证明该方法比经典阴影协议产生更精确的估计结果。同时，草图层析成像在可观测量估计中的准确性也优于通过最大似然估计公式训练的量子态。