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  • Open Access

Towards Robust Variational Quantum Simulation of Lindblad Dynamics via Stochastic Magnus Expansion

Jia-Cheng Huang1,*, Hao-En Li1,2,*, Yi-Cheng Wang1, Guang-Ze Zhang1, Jun Li1,3,4, and Han-Shi Hu1,†

  • *These authors contributed equally to this work.
  • †Contact author: hshu@mail.tsinghua.edu.cn
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PRX Quantum 6, 040312 – Published 17 October, 2025

DOI: https://doi.org/10.1103/yyln-q22s

Abstract

Dedicated to the centenary of quantum mechanics. In this paper, we introduce a novel and general framework for the variational quantum simulation of Lindblad equations. Building on the close relationship between the unraveled Lindblad dynamics, stochastic Magnus integrators, and variational quantum simulation, we propose a high-order scheme for solving the quantum state diffusion equation using exponential integrators. This formulation facilitates the simulation of wavefunction trajectories within the established framework of variational quantum algorithms for time evolution. Our algorithm significantly enhances robustness in two key aspects: the stability of the simulation with large time steps, and the reduction in the number of quantum trajectories required to accurately simulate the Lindblad dynamics in terms of the ensemble average. We demonstrate the effectiveness of our algorithm through numerical examples in both classical and quantum implementations, including the transverse-field Ising model (TFIM) with damping, the Fenna-Matthews-Olson (FMO) complex, and the radical pair model (RPM). The simulation accuracy can be systematically improved, and the algorithm remains reliable even in highly oscillatory regimes. These methods are expected to be applicable to a broader class of open quantum systems beyond the specific models considered in this study.

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Popular Summary

Quantum systems are extremely delicate and rarely exist in isolation. In everyday life, they are constantly influenced by their surroundings—gaining and losing energy in ways that shape how they behave. Understanding these effects is crucial not only for explaining natural phenomena, such as how plants harvest light during photosynthesis or how birds sense Earth’s magnetic field, but also for building reliable quantum technologies. The Lindblad equation is the standard way scientists describe these open quantum systems. Nonetheless, solving it on today’s quantum computers is very demanding compared to the Hamiltonian simulation.

In this work, we introduce a new method that makes such simulations more practical on near-term quantum devices. Our approach combines variational quantum algorithms with a mathematical tool called the stochastic Magnus expansion, making it especially well suited for current hardware. Our method overcomes the variance accumulation problem often faced by quantum trajectory approaches through high-order expansions, and its stability can be systematically improved. We tested it on several important examples from physics, chemistry, and biology, and found that it performs reliably—even for the avian compass, a notoriously challenging quantum system to simulate.

Our new algorithm bridges the gap between variational quantum simulations of the Lindblad equation and high-precision numerical solvers for stochastic dynamics. While tools for simulating open systems are still emerging, our approach may already be practical on today’s quantum devices and could open the door to broader studies of real-world quantum phenomena and future technologies.

Article Text

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PRX Quantum 6, 040312– Published 17 October, 2025
  • Received 28 March 2025
  • Revised 7 July 2025
  • Accepted 16 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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