A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation

A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation

9 Jul 2024 | Hongrui Chen, Bowen Li, Jianfeng Lu, and Lexing Ying
This paper introduces a randomized method for simulating Lindblad equations, which is particularly useful for quantum Gibbs sampling. The method involves randomly sampling a single Lindbladian at each time step, significantly simplifying the implementation compared to existing methods that require multiple jump operators. The approach is analyzed in two setups: the average channel and the random channel, with detailed convergence analyses provided for both. The average channel setup shows that the diamond distance error scales as \(O(T^2/M)\), while the random channel setup demonstrates that the average weighted \(\ell^2\) error scales as \(O(T/\sqrt{M})\). The method is applied to quantum Gibbs sampling, where it ensures fast thermalization for systems with spectral densities close to the semi-circle law. An example of the Davies generator, a class of quantum Markovian systems that thermalize to the Gibbs state, is constructed using the randomized method, showing efficient thermalization for random Hamiltonians. The paper concludes with future directions, including high-order methods for Lindbladian simulation and concentration-type analysis for the random channel.This paper introduces a randomized method for simulating Lindblad equations, which is particularly useful for quantum Gibbs sampling. The method involves randomly sampling a single Lindbladian at each time step, significantly simplifying the implementation compared to existing methods that require multiple jump operators. The approach is analyzed in two setups: the average channel and the random channel, with detailed convergence analyses provided for both. The average channel setup shows that the diamond distance error scales as \(O(T^2/M)\), while the random channel setup demonstrates that the average weighted \(\ell^2\) error scales as \(O(T/\sqrt{M})\). The method is applied to quantum Gibbs sampling, where it ensures fast thermalization for systems with spectral densities close to the semi-circle law. An example of the Davies generator, a class of quantum Markovian systems that thermalize to the Gibbs state, is constructed using the randomized method, showing efficient thermalization for random Hamiltonians. The paper concludes with future directions, including high-order methods for Lindbladian simulation and concentration-type analysis for the random channel.
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