A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion
Abstract
Developing general-purpose numerical methods for non-Markovian quantum state diffusion has long been a challenge in the simulation of open quantum systems. This paper proposes a deterministic polynomial chaos Galerkin method, which is easy to implement, accurate, and efficient. By combining a low-rank decomposition of bath correlations with Galerkin projection onto a polynomial chaos basis, the method yields a deterministic formulation compatible with standard numerical solvers for partial differential equations. The reduced density matrix is obtained from a single deterministic solve at a cost comparable to that of computing one stochastic hierarchy trajectory. Replacing an ensemble of Ntraj trajectories therefore reduces the computational cost by approximately a factor of Ntraj and eliminates statistical sampling error. Benchmark comparisons show that the proposed method achieves smaller errors at lower computational cost than the tested stochastic hierarchy methods. Applications involving nonquadratic potentials, diffraction and interference, and coupled three-dimensional dynamics demonstrate its ability to handle challenging problems.
Related
- A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems
- Physical-Field Reconstruction from Sparse Observations: When Are Diffusion Models Preferable to Deterministic Regression?
- Non-Markovian delay-assisted sensing with waveguide-coupled quantum emitters
- Quantum Simulation of Markovian and Non-Markovian Open Quantum Dynamics in Heavy-Ion Collisions
- Open-quantum-system theory of non-Markovian electron-phonon dynamics
BibTeX
@misc{zhu2026deterministic,
title = {{A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion}},
author = {Quanhui Zhu and Zhenning Cai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12070},
eprint = {2610.12070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12070},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion
AU - Quanhui Zhu
AU - Zhenning Cai
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12070
UR - https://arxiv.org/abs/2610.12070
AB - Developing general-purpose numerical methods for non-Markovian quantum state diffusion has long been a challenge in the simulation of open quantum systems. This paper proposes a deterministic polynomial chaos Galerkin method, which is easy to implement, accurate, and efficient. By combining a low-rank decomposition of bath correlations with Galerkin projection onto a polynomial chaos basis, the method yields a deterministic formulation compatible with standard numerical solvers for partial differential equations. The reduced density matrix is obtained from a single deterministic solve at a cost comparable to that of computing one stochastic hierarchy trajectory. Replacing an ensemble of N_traj trajectories therefore reduces the computational cost by approximately a factor of N_traj and eliminates statistical sampling error. Benchmark comparisons show that the proposed method achieves smaller errors at lower computational cost than the tested stochastic hierarchy methods. Applications involving nonquadratic potentials, diffraction and interference, and coupled three-dimensional dynamics demonstrate its ability to handle challenging problems.
ER -