Quantum algorithms from fluctuation theorems: Thermal-state preparation

Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and work distribution arising non-equilibrium process that connects two with Hamiltonians $H_0$ $H_1=H_0+V$. Building upon these theorems, we present algorithm to prepare purification the state $H_1$ at inverse temperature $\beta \ge 0$ starting from $H_0$. The complexity algorithm, given by number uses certain unitaries, is $\tilde {\cal O}(e^{\beta (\Delta \! A- w_l)/2})$, where $\Delta A$ free-energy difference $H_0,$ $w_l$ cutoff depends on approximation error $\epsilon>0$. If trivial, this exponential \|V\|$, $\|V\|$ spectral norm $V$. This represents significant improvement prior algorithms have \|H_1\|$ regime $\|V\|\ll \|H_1\|$. dependence $\epsilon$ varies according structure systems. It can be $1/\epsilon$ general, but show it sublinear if commute, or polynomial are local spin possibility applying unitary drives system out allows one increase value improve even further. To end, analyze for preparing transverse field Ising model using different processes see improvements.