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 H0 and H1 = H0 + V . Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H1 at inverse temperature β ≥ 0 starting from a purification of the thermal state of H0 at the same temperature. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is Õ(eβ(∆A−wl)/2), where ∆A is the free-energy difference between the two quantum systems and wl is a work cutoff that depends on the properties of the work distribution and the approximation error ε > 0. If the non-equilibrium process is trivial, this complexity is exponential in β∥V∥, where ∥V∥ is the spectral norm of V . This represents a significant improvement over prior quantum algorithms that have complexity exponential in β∥H1∥ in the regime where ∥V∥ ≪ ∥H1∥. 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 H0 but not for preparing the thermal state of H1. This can occur, for example, when H0 is an integrable quantum system and V introduces interactions such that H1 is non-integrable. The dependence of the complexity in ε, when all other parameters are fixed, varies according to the structure of the quantum systems. It can be exponential in 1/ε in general, but we show it to be sublinear in 1/ε if H0 and H1 commute, or polynomial in 1/ε if H0 and H1 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 wl 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.
Original language | English |
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Article number | 825 |
Pages (from-to) | 1-55 |
Number of pages | 55 |
Journal | Quantum |
Volume | 6 |
DOIs | |
Publication status | Published - 6 Oct 2022 |
Externally published | Yes |
Bibliographical note
Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.Keywords
- Quantum algorithms
- Thermodynamics