Nonequilibrium fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and work distributions arising in nonequilibrium processes. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of a quantum system of interest. Unlike previous algorithms based on a thermalization process that brings an infinite-temperature state to one at finite temperature, our algorithm assumes access to the purification of the thermal state of $H_0$ to prepare a purification of the thermal state of $H_1=H_0+V$ at the same temperature. When the perturbation $V$ is small, even with a trivial nonequilibrium process our algorithm provides a significant improvement over prior quantum algorithms in terms of complexity by exploiting the similarity between the two thermal states. Further improvements arise from a judicious choice of the nonequilibrium process.
I will start the talk with a review of fluctuation theorems relevant to this work before describing the thermal state preparation algorithm in detail. The essential ingredients of our algorithm are the definition of a "work operator", introduction of a work cutoff and an approximation of the exponentiated work operator on a subspace of work values above the cutoff. Special cases of when $H_0$ and $H_1$ commute and when they are both local spin Hamiltonians will be discussed. For the transverse field Ising model I will numerically demonstrate the effect of using different nonequilibrium processes on the complexity of the algorithm.
