Approximation of Free Energies with Fluctuation Relations on Quantum Hardware*

Fluctuation relations allow for the computation of equilibrium properties, like free energy differences, from an ensemble of non-equilibrium dynamics simulations. Computing them for quantum systems is exponentially hard on classical computers because it requires taking the exponential of an exponentially scaling Hamiltonian. Given that quantum computers can alleviate this hurdle, we propose an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We prove that the approximation is rigorously an upper bound for the free energy difference and that the computation is exact when the inverse temperature goes to zero or infinity in adiabatic regimes. Additionally, a rigorous bound is given for the non-adiabatic regime. Furthermore, we successfully demonstrate a proof-of-concept of our algorithm using the transverse field Ising model on quantum hardware. Since free energies play a critical role in any equilibrium property, our algorithm may eventually serve as a valuable tool in a wide range of applications including the construction of phase diagrams, prediction of transport properties and reaction constants, and computer-aided drug design in the future.