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arXiv:2512.14015

Frozen Gaussian sampling algorithms for simulating Markovian open quantum systems in the semiclassical regime

Limin Xu, Zhen Huang, Zhennan Zhou

quant-phmath.NASubmitted Dec 16, 2025• Announced Dec 17, 2025• 0 citations
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Abstract

Simulating Markovian open quantum systems in the semiclassical regime poses a grand challenge for computational physics, as the highly oscillatory nature of the dynamics imposes prohibitive resolution requirements on traditional grid-based methods. To overcome this barrier, this paper introduces an efficient Frozen Gaussian Sampling (FGS) algorithm based on the Wigner-Fokker-Planck phase-space formulation. The proposed algorithm exhibits two transformative advantages. First, for the computation of physical observables, its sampling error is independent of the semiclassical parameter $\varepsilon$, thus fundamentally breaking the prohibitive computational scaling faced by grid methods in the semiclassical limit. Second, its mesh-free nature entirely eliminates the boundary-induced instabilities that constrain long-time grid-based simulations. Leveraging these capabilities, the FGS algorithm serves as a powerful investigatory tool for exploring the long-time behavior of open quantum systems. Specifically, we provide compelling numerical evidence for the existence of steady states in strongly non-harmonic potentials-a regime where rigorous analytical results are currently lacking.

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Simulating Markovian open quantum systems in the semiclassical regime poses a grand challenge for computational physics, as the highly oscillatory nature of the dynamics imposes prohibitive resolution requirements on traditional grid-based methods. To overcome this barrier, this paper introduces an efficient Frozen Gaussian Sampling (FGS) algorithm based on the Wigner-Fokker-Planck phase-space formulation. The proposed algorithm exhibits two transformative advantages. First, for the computation of physical observables, its sampling error is independent of the semiclassical parameter ε\varepsilonε, thus fundamentally breaking the prohibitive computational scaling faced by grid methods in the semiclassical limit. Second, its mesh-free nature entirely eliminates the boundary-induced instabilities that constrain long-time grid-based simulations. Leveraging these capabilities, the FGS algorithm serves as a powerful investigatory tool for exploring the long-time behavior of open quantum systems. Specifically, we provide compelling numerical evidence for the existence of steady states in strongly non-harmonic potentials-a regime where rigorous analytical results are currently lacking.
Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. A common approach to the analysis of such systems is to approximate the non-Markovian environment by discrete bosonic modes thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been proposed in previous works [D. Tamascelli et al, PRL (2012), A. W. Chin et al, J. of Math. Phys (2010)], the resulting approximation lacks rigorous convergence guarantees. In this paper, we initiate a rigorous study of the convergence properties of these methods. We show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that, for most physically interesting models of non-Markovian environments, the approximation error falls off polynomially with the number of modes. Our results lend rigor to numerical methods used for approximating non-Markovian quantum dynamics and allow for a quantitative assessment of classical as well as quantum algorithms in simulating non-Markovian quantum systems.
One of the main challenges in numerical simulation of quantum dynamics is the prohibitive cost in the semi-classical regime, in which the de Broglie wave length is small compared with the characteristic length scale and the solution is highly oscillatory. For the von-Neumann equation for mixed-state quantum dynamics, this difficulty is overcome by using the Weyl variables, under which the solution is not oscillatory. Furthermore, we use the integral representation of the potential difference, which robustly captures the classical limit as the semi-classical parameter approaches zero. By using exact Hermite moments and quantum singular value transformation to treat the polynomial and sparse coordinate matrices of the dense projected smooth non-polynomial potentials respectively, we obtain a quantum algorithm efficient for {\it all} ranges of wave lengths, with complexity {\it polynomial} in the spatial dimension and discretization and query bounds containing {\it no} negative powers of the small wavelength. Thus it can capture the correct physical observables even if the spatial grid does not resolve the frequency, hence defying the Nyquist-Shannon sampling theorem.
Simulating phase transitions using quantum computers presents a significant challenge due to the complex and often chaotic nature of these transitions. Traditional quantum simulation methods often rely on uniform sampling of the relevant phase space, leading to inefficient resource utilization and prohibitively long simulation times. This work proposes a novel approach to quantum simulation of phase transitions that leverages adaptive sampling, guided by reinforcement learning, to dynamically adjust the sampling strategy. The core idea is to identify and focus on regions of the phase space where the transition is most pronounced, thereby minimizing the overall computational cost. We formulate the problem as a Markov decision process, where the quantum computer's sampling strategy is the agent and the simulation's performance is the reward. Reinforcement learning algorithms are employed to train the agent, enabling it to learn an optimal sampling policy. We demonstrate, through theoretical analysis and conceptual design, the potential of this adaptive sampling method to significantly reduce the computational burden associated with simulating phase transitions while maintaining accuracy. Future work will involve exploring specific algorithms and architectures for implementing this approach on actual quantum hardware.
We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024
When simulating the dynamics of open quantum systems with quantum computers, it is essential to accurately approximate the system's behaviour while preserving the physicality of its evolution. Traditionally, for Markovian open quantum systems, this has been achieved using first and second-order Trotter-Suzuki product formulas or probabilistic algorithms. In this work, we introduce novel non-probabilistic algorithms for simulating Markovian open quantum systems using randomisation. Our methods, including first and second-order randomised Trotter-Suzuki formulas and the QDRIFT channel, not only maintain the physicality of the system's evolution but also enhance the scalability and precision of quantum simulations. We derive error bounds and step count limits for these techniques, bypassing the need for the mixing lemma typically employed in Hamiltonian simulation proofs. Furthermore, we implement these randomised algorithms using Classical Sampling (CS), demonstrating their gate complexity advantages over deterministic TS product formulas. This work systematically extends powerful randomisation techniques from Hamiltonian simulation to the general setting of Markovian open quantum systems, highlighting their potential to enable faster and more accurate simulations.
While several numerical techniques are available for predicting the dynamics of non-Markovian open quantum systems, most struggle with simulations for very long memory and propagation times, e.g., due to superlinear scaling with the number of time steps n . Here, we introduce a numerically exact algorithm to calculate process tensors—compact representations of environmental influences—which provides a scaling advantage over previous algorithms by leveraging self-similarity of the tensor networks that represent the environment. It is applicable to environments with Gaussian statistics, such as for spin-boson-type open quantum systems. Based on a divide-and-conquer strategy, our approach requires only O(n log n) singular value decompositions for environments with infinite memory. Where the memory can be truncated after nc time steps, a nominal scaling O(nc log nc) is found, which is independent of n . This improved scaling is enabled by identifying process tensors with repeatable blocks. To demonstrate the power and utility of our approach, we provide three examples. (1) We calculate the fluorescence spectra of a quantum dot under both strong driving and strong dot-phonon couplings, a task requiring simulations over millions of time steps, which we are able to perform in minutes. (2) We efficiently find process tensors describing superradiance of multiple emitters. (3) We explore the limits of our algorithm by considering coherence decay with a very strongly coupled environment. The observed computation time is not necessarily proportional to the number of singular value decompositions because the matrix dimensions also depend on the number of time steps. Nevertheless, quasilinear and sublinear scaling of computation time is found in practice for a wide range of parameters. While there are instances where existing methods can achieve comparable nominal scaling by precalculating effective propagators for time-independent or periodic system Hamiltonians, process tensors contain all the information needed to extract arbitrary multitime correlation functions of the system when driven by arbitrary time-dependent system Hamiltonians. The algorithm we present here not only significantly extends the scope of numerically exact techniques to open quantum systems with long memory times, but it also has fundamental implications for the simulation complexity of tensor network approaches. Published by the American Physical Society 2024