Quantum algorithms from fluctuation theorems: Thermal-state preparation

Abstract

Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with HamiltoniansH0andH1=H0+V. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state ofH1at inverse temperatureβ≥0starting from a purification of the thermal state ofH0. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, isO~(eβ(ΔA−wl)/2), whereΔAis the free-energy difference betweenH1andH0,andwlis a work cutoff that depends on the properties of the work distribution and the approximation errorϵ>0. If the non-equilibrium process is trivial, this complexity is exponential inβ‖V‖, where‖V‖is the spectral norm ofV. This represents a significant improvement of prior quantum algorithms that have complexity exponential inβ‖H1‖in the regime where‖V‖≪‖H1‖. The dependence of the complexity inϵvaries according to the structure of the quantum systems. It can be exponential in1/ϵin general, but we show it to be sublinear in1/ϵifH0andH1commute, or polynomial in1/ϵifH0andH1are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value ofwland improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.