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 $N_{\mathrm{traj}}$ trajectories therefore reduces the computational cost by approximately a factor of $N_{\mathrm{traj}}$ 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.
Keywords
Publication details
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Cite this article
APA 7
Zhu, Q., & Cai, Z. (2026). A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion. https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion
MLA 9
Zhu, Quanhui, and Zhenning Cai. "A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion." https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion.
Chicago (author–date)
Zhu, Quanhui, and Zhenning Cai. 2026. "A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion." https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion.
Harvard
Zhu, Q. and Cai, Z. (2026) 'A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion', Available at: https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion.
Vancouver
Zhu Q, Cai Z. A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion. https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion
IEEE
Q. Zhu, and Z. Cai, "A deterministic polynomial chaos Galerkin method for non-Markovian quantum state diffusion," https://omanscience.com/en/articles/a-deterministic-polynomial-chaos-galerkin-method-for-non-markovian-quantum-state-diffusion.