Energy-filtered random-phase states as microcanonical thermal pure quantum states

We propose a method to calculate finite-temperature properties of many-body systems for microcanonical ensembles, which may find a potential application of near-term quantum computers. In our formalism, a microcanonical ensemble is specified with a target energy and a width of the energy window, by expressing the density of states as a sum of Gaussians centered at the target energy with its spread associated with the width of the energy window. Using the Fourier representation of the Gaussian, we then show that thermodynamic quantities such as entropy and temperature can be calculated by evaluating the trace of the time-evolution operator, and the trace of the time-evolution operator multiplied by the Hamiltonian of the system. We also describe how these traces can be evaluated using random diagonal-unitary circuits suitable for quantum computation. We demonstrate the proposed method by numerically calculating thermodynamic quantities of the one-dimensional spin-1/2 Heisenberg model on small clusters and show that the proposed method is most effective for the target energy around which a larger number of energy eigenstates exist.