Learning shadows to predict quantum ground state correlations

TL;DR

This paper presents a novel variational method inspired by shadow tomography to efficiently approximate quantum ground state correlations, enabling better predictions of local observables and correlations in quantum spin systems.

Contribution

The authors introduce a shadow tomography-inspired variational scheme that uses a bag of parametrized snapshots to estimate ground state correlations more comprehensively than traditional density matrix methods.

Findings

Method efficiently predicts ground state energy and correlations.

Numerical results show the approach is parallelizable and computationally feasible.

The scheme captures correlations beyond reduced density matrices.

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. Unlike reduced density matrix approaches, learning the underlying distribution of measurement outcomes allows one to further correlations beyond those in the constrained density matrix. We show, with numerical results, that the proposed variational method can be parallelized, is efficiently simulable, and yields a more complete description of the ground state.