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We introduce two kinds of quantum algorithms to explore microcanonical andcanonical properties of many-body systems. The first one is a hybrid quantumalgorithm that, given an efficiently preparable state, computes expectationvalues in a finite energy interval around its mean energy. This algorithm isbased on a filtering operator, similar to quantum phase estimation, whichprojects out energies outside the desired energy interval. However, instead ofperforming this operation on a physical state, it recovers the physical valuesby performing interferometric measurements without the need to prepare thefiltered state. We show that the computational time scales polynomially withthe number of qubits, the inverse of the prescribed variance, and the inverseerror. In practice, the algorithm does not require the evolution for longtimes, but instead a significant number of measurements in order to obtainsensible results. Our second algorithm is a quantum-assisted Monte Carlosampling method to compute other quantities which approach the expectationvalues for the microcanonical and canonical ensembles. Using classical MonteCarlo techniques and the quantum computer as a resource, this methodcircumvents the sign problem that is plaguing classical Quantum Monte Carlosimulations, as long as one can prepare states with suitable energies. Allalgorithms can be used with small quantum computers and analog quantumsimulators, as long as they can perform the interferometric measurements. Wealso show that this last task can be greatly simplified at the expense ofperforming more measurements.

Algorithms for quantum simulation at finite energies