Learning shadows to predict quantum ground state correlations

Abstract

We introduce a variational scheme inspired by classical shadow tomography to compute ground state correlations of quantum spin Hamiltonians. Shadow tomography allows for efficient reconstruction of expectation values of arbitrary observables from a bag of repeated, randomized measurements, called snapshots, on copies of the state $\rho$. The prescription allows one to infer expectation values of $M$ $k-$local observables to accuracy $\epsilon$ using just $N \sim 3^k \text{log}M /\epsilon^2$ snapshots when measurements are performed in locally random bases. Turning this around, a bag of snapshots can be considered an efficient representation of the state $\rho$, particularly for estimating low-weight observables, such as terms in a local Hamiltonian needed to estimate the energy. Inspired by this, we consider a variational scheme wherein a bag of $N$ parametrized snapshots is used to represent the putative ground state of a desired local spin Hamiltonian and optimized to lower the energy with respect to it. Additional constraints in the form of positivity of reduced density matrices, motivated by work in quantum chemistry, are employed to ensure compatibility of the predicted correlations with the underlying Hilbert space. Despite the limited set of imposed constraints, we show that the optimized ensemble of snapshots yields fairly accurate estimates of the ground state energy and other correlations, including those that do not appear in the set of constraints used in optimization.