Algorithms for Quantum Simulation at Finite Energies

Abstract

We consider a quantum algorithm to compute expectation values of observables in a finite energy interval for many-body problems. It is based on a filtering operator, similar to quantum phase estimation, which projects out energies outside that interval. However, instead of performing this operation on a physical state, it recovers the physical values by performing interferometric measurements without the need to prepare the filtered state. We show that the computational time scales polynomially with the number of qubits, the inverse of the prescribed variance, and the inverse error. In practice, the algorithm does not require the evolution for long times, but instead a significant number of measurements in order to obtain sensible results. We then propose a hybrid classical-quantum algorithm to compute other quantities which approach the expectation values for the microcanonical and canonical ensembles. They utilize classical Monte Carlo techniques, where the sampling algorithms use the quantum computer as a resource. All algorithms can be used with small quantum computers and analog quantum simulators, as long as they can perform the interferometric measurements. We also show that this last task can be greatly simplified at the expense of performing more measurements.