Quantum algorithms from fluctuation theorems: Thermal-state preparation

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 H-0 and H-1 = H-0+V. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H-1 at inverse temperature beta >= 0 starting from a purification of the thermal state of H-0 at the same temperature. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O tilde (e(beta(degrees`4-wl)/2)), where delta A is the free-energy difference between the two quantum systems and w(l) is a work cutoff that depends on the properties of the work distribution and the approximation error is an element of > 0. If the non-equilibrium process is trivial, this complexity is exponential in beta IIVII, where IIVII is the spectral norm of V. This represents a significant improvement over prior quantum algorithms that have complexity exponential in beta IIH1II in the regime where IIVII << IIH1II. The quantum algorithm is then expected to be advantageous in a setting where an efficient quantum circuit is available for preparing the purification of the thermal state of H-0 but not for preparing the thermal state of H-1. This can occur, for example, when H0 is an integrable quantum system and V introduces interactions such that H-1 is non-integrable. The dependence of the complexity in is an element of, when all other parameters are fixed, varies according to the structure of the quantum systems. It can be exponential in 1/is an element of in general, but we show it to be sublinear in 1/is an element of if H-0 and H-1 commute, or polynomial in 1/is an element of if H-0 and H-1 are local spin systems. In addition, 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.