Variational Microcanonical Estimator

An important target for noisy intermediate-scale quantum computers is to calculate thermal properties of complex interacting quantum systems. We propose a variational quantum algorithm for estimating microcanonical expectation values in a nonintegrable model. Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density. These states can in turn be used to estimate microcanonical averages of local operators via the eigenstate thermalization hypothesis (ETH) in combination with classical averaging over a sufficiently large ensemble of variational states. We apply the algorithm to the one-dimensional mixed field Ising model, where it converges for ansatz circuits of roughly linear depth in system size. This allows us to present results for up to 13 qubits. The most accurate thermal estimates are produced at intermediate energy density, the range usually considered to be the most difficult to capture. We also connect our problem to recent works investigating the underpinnings of the ETH, in particular the pseudo-random nature of off-diagonal matrix elements of local operators in highly excited energy eigenstates.