A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation

Abstract

We study a qDRIFT-type randomized method to simulate Lindblad dynamics by decomposing its generator into an ensemble of Lindbladians, $\mathcal{L}={\sum }_{a\in \mathcal{A}}{\mathcal{L}}_a$, where each ${\mathcal{L}}_a$ comprises a simple Hamiltonian and a single jump operator. Assuming an efficient quantum simulation is available for the Lindblad evolution $e^{t{\mathcal{L}}_a}$, we implement $e^{t{\mathcal{L}}_a}$ for a randomly sampled ${\mathcal{L}}_a$ at each time step according to a probability distribution $\mu$ over the ensemble $\{{\mathcal{L}}_a{\}}_{a\in \mathcal{A}}$. This randomized strategy reduces the quantum cost of simulating Lindblad dynamics, particularly in quantum many-body systems with a large or even infinite number of jump operators. Our contributions are two-fold. First, we provide a detailed convergence analysis of the proposed randomized method, covering both average and typical algorithmic realizations. This analysis extends the known results for the random product formula from closed systems to open systems, ensuring rigorous performance guarantees. Second, based on the random product approximation, we derive a new quantum Gibbs sampler algorithm that utilizes jump operators sampled from a Clifford-random circuit. This generator (i) can be efficiently implemented using our randomized algorithm, and (ii) exhibits a spectral gap lower bound that depends on the spectrum of the Hamiltonian. Our results present a new instance of a class of Hamiltonians for which the thermal states can be efficiently prepared using a quantum Gibbs sampling algorithm.

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