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 Hamiltonians H₀ and H₁=H₀+V. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H₁ at inverse temperature β ≥ 0 starting from a purification of the thermal state of H₀. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O(e^β (Δ A- w_l)/2), where Δ A is the free-energy difference between H₁ and H₀, and w_l is a work cutoff that depends on the properties of the work distribution and the approximation error ε>0. If the non-equilibrium process is trivial, this complexity is exponential in β V, where V is the spectral norm of V. This represents a significant improvement of prior quantum algorithms that have complexity exponential in β H₁ in the regime where V≪ H₁. The dependence of the complexity in ε varies according to the structure of the quantum systems. It can be exponential in 1/ε in general, but we show it to be sublinear in 1/ε if H₀ and H₁ commute, or polynomial in 1/ε if H₀ and H₁ are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of w_l and 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.