Efficient Classical Shadow Tomography with Matrix Product States*

Classical shadow tomography with locally scrambled quantum dynamics can predict many properties of quantum states with very few measurements using shallow circuits. However, it is unknown whether there exists an algorithm that can efficiently process the classical shadow data and make predictions of the properties in polynomial time. Here, we design an efficient, polynomial time algorithm based on tensor network representations of the reconstruction channel to make predictions on various properties, such as quantum fidelity, and expectation of Pauli observables. Then, we demonstrate this method gives unbiased predictions using large system-sizes numerics. In addition, we also discuss sample complexity in the shallow-circuit regime and the sensitivity due to different system parameters. Our work shed lights on how classical shadow tomography can be efficiently utilized to learn state information on near-term quantum devices.