Tensor Network Tomography using Classical Shadows

Classical shadow tomography is a powerful tool for extracting large amounts of physically relevant information from a quantum state using a polynomial number of measurements. However, classical shadows are not quantum states — they are not completely positive, and thus are limited to the estimation of expectation values. Intermediate representations of states derived from shadows that are completely positive but retain the computational benefits of polynomial scaling in estimation are a valuable tool, if they can be constructed with only polynomial (in qubit number) classical computation. Here we present theoretical and numerical progress toward constructing tensor network state ansatzes from classical shadows data. Our starting point is an existing protocol by Baumgratz, Gross, Cramer, and Plenio for reconstructing Matrix Product States (MPSs) from local correlation functions. We implement this protocol with data from classical shadows. We also go beyond the protocol of Plenio et al. by investigating the use of shadow data to reconstruct the locality structure of system, after which the MPS tomography algorithm can be applied. We discuss prospects for extending this technique to different tensor network ansatzes, such as PEPS or MERA, and to higher dimensions.