## 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 β ≥ 0 starting from a purification of the thermal state of H_{0} at the same temperature. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is Õ(e^{β}(∆^{A−wl}^{)}/^{2}), where ∆A is the free-energy difference between the two quantum systems and w_{l} 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∥H_{1}∥ in the regime where ∥V∥ ≪ ∥H_{1}∥. 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 H_{0} but not for preparing the thermal state of H_{1}. This can occur, for example, when H_{0} is an integrable quantum system and V introduces interactions such that H_{1} 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 H_{0} and H_{1} commute, or polynomial in 1/ε if H_{0} and H_{1} 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 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.

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