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 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 $\beta \ge 0$ starting from a purification of the thermal state of $H_0$. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is $\tilde{\mathcal{O}}(e^{\beta (\Delta A-w_l)/2})$, where $\Delta A$ is the free-energy difference between $H_1$ and $H_0,$ and $w_l$ is a work cutoff that depends on the properties of the work distribution and the approximation error $\epsilon >0$. If the non-equilibrium process is trivial, this complexity is exponential in $\beta \Vert V\Vert$, where $\Vert V\Vert$ is the spectral norm of $V$. This represents a significant improvement of prior quantum algorithms that have complexity exponential in $\beta \Vert H_1\Vert$ in the regime where $\Vert V\Vert \ll \Vert H_1\Vert$. The dependence of the complexity in $\epsilon$ varies according to the structure of the quantum systems. It can be exponential in $1/\epsilon$ in general, but we show it to be sublinear in $1/\epsilon$ if $H_0$ and $H_1$ commute, or polynomial in $1/\epsilon$ if $H_0$ and $H_1$ 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.

arXiv comment54 pages, 11 figures

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