Southwest Quantum Information and Technology

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Presenting Author: Burak Sahinoglu, PsiQuantum
Contributing Author(s): Zoe Holmes, Gopikrishnan Muraleedharan, Rolando D. Somma, Yigit Subasi

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 \(\epsilon>0\). 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.

Read this article online: https://arxiv.org/abs/2203.08882

(Session 4 : Thursday from 4:15 pm - 4:45 pm)