Quantum algorithms from fluctuation theorems: Thermal-state preparation

Zoe Holmes1, Gopikrishnan Muraleedharan2, Rolando D. Somma2, Yigit Subasi1, and Burak Şahinoğlu2

1Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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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 $\tilde {\cal O}(e^{\beta (\Delta \! A- w_l)/2})$, 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 $\epsilon\gt0$. If the non-equilibrium process is trivial, this complexity is exponential in $\beta \|V\|$, where $\|V\|$ is the spectral norm of $V$. This represents a significant improvement of prior quantum algorithms that have complexity exponential in $\beta \|H_1\|$ in the regime where $\|V\|\ll \|H_1\|$. 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.

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