Scalable Learning Algorithms based on Shadow Tomography

Abstract

Classical shadow tomography has been shown to be the optimal protocol for estimating linear properties of quantum states using independent single-copy measurements. Efforts have been made to improve the sample complexity of the protocol by finding experi- mentally feasible and mathematically tractable choices for the unitary ensemble used for measurements. One such approach utilizes locally scrambled unitary ensembles which ex- hibit a clean analytical form for the reconstruction map enabling us to construct shadows with superior sample complexity. On a related front, there has been a surge in develop- ing learning algorithms for quantum states based on measurement data. Tensor networks present themselves as a natural choice for the learning ansatz due to their efficient rep- resentation of low-entanglement states and easy manipulation. They are also uniquely compatible with the tomography protocol based on locally scrambled ensembles. The main achievement of this work is a new learning algorithm that couples locally scram- bled shadow tomography with stochastic optimization techniques on a manifold to learn a purified MPS representation of quantum states. We also couple pre-existing learning algorithms with locally scramble shadows and present a general study of these algorithms in the language of learning theory. Finally, we describe the pivotal factors that should be considered while designing these algorithms.