Shadow learning for efficiently estimating ground state properties

Shadow tomography describes an efficient protocol to measure quantum states in randomized bases and a postprocessing scheme that uses this classical data to infer k-local properties of the state without having to learn the entire density matrix of the system. We consider an inversion of this approach, which we dub "shadow learning", in which a bag of snapshots, or classical measurement data is guessed, and then variationally optimized to obtain the k-local correlations in the ground state of Hamiltonians of interest. The cost of such computation in general scales only polynomially in system size and can be easily parallelized further to reduce compute time signifcantly; these methods are also not restricted to one-dimensional systems. This is a reflection of that the fact that we forgo any attempt at obtaining arbitrary properties of the ground state, and instead only focus on obtaining relevant correlations that characterize important physical features and which are generally low weight. Despite the potential allure of the approach in terms of scaling, the bag of snapshots in general produce unphysical expectational values of correlations unless constraints are imposed and therefore cannot be naively optimized to minimize the energy of the state with respect to the Hamiltonian. We discuss in detail the set of constraints that appear to provide accurate results for the ground state properties of trial Hamiltonians.