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arXiv:2610.36257

Dynamical Tensor Train Approximation for Wigner--Fokker--Planck Equation with Applications to Open Quantum Systems

Geshuo Wang, Jingwei Hu

math.NAquant-phSubmitted Oct 26, 2026• Announced Oct 28, 2026• 0 citations
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Abstract

We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.

AI Summary

The paper addresses deterministic simulation of the Wigner–Fokker–Planck equation for continuous-variable open quantum systems, where a $d$-dimensional configuration space produces a $2d$-dimensional phase-space problem. It combines tensor-train (TT) compression of the Wigner function with dynamical low-rank evolution to avoid storing the full phase-space grid. Each position and momentum coordinate is a separate TT mode. A fixed-rank projector-splitting integrator advances the state through local core and bond updates. The nonlocal Wigner operator uses a CP or TT representation of the potential difference, momentum-space FFTs, and matrix-free exponential actions; transport, friction, and diffusion use finite differences and forward Euler updates. For bounded potential-representation complexity and moderate exponential-action iteration counts, the per-step cost is cubic in the solution rank $r$, linear in spatial resolution, and quasi-linear in momentum resolution. Storage scales as $\mathcal{O}(dr^2(N_x+N_p))$. Numerical tests include comparisons with existing one- and two-dimensional examples, rank-refinement tests, and a three-dimensional open-system calculation on a $64^6$ grid. Rank $24$ requires 150,528 stored entries instead of approximately $6.87\times10^{10}$. Dissipation reduces effective ranks in the tested weakly nonquadratic example. The six-dimensional test shows qualitative time-step consistency and small mass errors. The method assumes compressibility of both the state and potential difference. It uses fixed ranks and first-order time integration, and does not enforce mass conservation or complete positivity. General rank reduction under dissipation is not established; rank adaptivity, higher-order integration, and structure preservation remain future directions.

Little sign of AI-generated textIncrementalMissing comparisonStrong assumptions

Central claim may not hold: The demonstrated preprocessing is not entirely free of full tensor-product computations: the 2D2P benchmark constructs the full potential-difference tensor for TT-SVD and reconstructs tensors for approximation-error checks. The six-dimensional example demonstrates compressed simulation and timing scalability but does not quantitatively establish solution accuracy or convergence with rank and grid resolution. The stated right-environment definition requires projected products L C R^T, whereas the paper uses L C R, giving generally incorrect local evolution operators unless an unstated alternative convention is used.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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35 pages, 8 figures

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math.NACross-listedarXiv:2606.24190•Jun 24, 2026
We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.
The paper addresses deterministic simulation of the Wigner–Fokker–Planck equation for continuous-variable open quantum systems, where a ddd-dimensional configuration space produces a 2d2d2d-dimensional phase-space problem. It combines tensor-train (TT) compression of the Wigner function with dynamical low-rank evolution to avoid storing the full phase-space grid.
Each position and momentum coordinate is a separate TT mode. A fixed-rank projector-splitting integrator advances the state through local core and bond updates. The nonlocal Wigner operator uses a CP or TT representation of the potential difference, momentum-space FFTs, and matrix-free exponential actions; transport, friction, and diffusion use finite differences and forward Euler updates. For bounded potential-representation complexity and moderate exponential-action iteration counts, the per-step cost is cubic in the solution rank rrr, linear in spatial resolution, and quasi-linear in momentum resolution. Storage scales as O(dr2(Nx+Np))\mathcal{O}(dr^2(N_x+N_p))O(dr2(Nx​+Np​)).
Numerical tests include comparisons with existing one- and two-dimensional examples, rank-refinement tests, and a three-dimensional open-system calculation on a 64664^6646 grid. Rank 242424 requires 150,528 stored entries instead of approximately 6.87×10106.87\times10^{10}6.87×1010. Dissipation reduces effective ranks in the tested weakly nonquadratic example. The six-dimensional test shows qualitative time-step consistency and small mass errors.
The method assumes compressibility of both the state and potential difference. It uses fixed ranks and first-order time integration, and does not enforce mass conservation or complete positivity. General rank reduction under dissipation is not established; rank adaptivity, higher-order integration, and structure preservation remain future directions.
Abstract. A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included. Key words. Tensor train, matrix product state, low-rank approximation, time-varying tensors, tensor differential equations, splitting integrator. AMS subject classifications. 15A18,15A69,65F99,65L05 1. Introduction. In
We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S 1 / S 2 interconversion dynamics of pyrazine after UV photoexcitation to the S 2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations.
A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train/matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation, and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.
We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and runtime barriers that limit conventional finite-difference and spectral schemes. Two complementary algorithms are developed: an imaginary-time projector that drives the condensate toward its variational ground state and a rank-adapted fourth-order Runge-Kutta integrator for real-time dynamics. The framework captures a broad range of physical scenarios - including barrier-confined condensates, quasi-random potentials, long-range dipolar interactions, and multicomponent spinor dynamics - without leaving the compressed representation. Relative to standard discretizations, the QTT approach achieves an exponential reduction in computational resources while retaining quantitative accuracy, thereby extending the practicable regime of Gross-Pitaevskii simulations on classical hardware. These results position tensor networks as a practical bridge between high-performance classical computing and prospective quantum hardware for the numerical treatment of nonlinear Schrodinger-type partial differential equations.
Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.
Recent advances in dynamical low-rank approximation (DLRA) have demonstrated its effectiveness in high-dimensional simulations. However, existing DLRA algorithms still face significant challenges when handling systems that involve complex collision terms, including the pseudo-differential operator (Ψ{\rm Ψ}Ψ) in the Wigner equation, a representative operator characterized by nonlocality. It is deserving to carry out a series of works to develop the DLRA algorithms for solving the Wigner equation. As the first step in this series of works, we propose an efficient DLRA algorithm for the Wigner equation, using a separable decomposition of the difference potential. We combine this separable assumption with two often-used truncations of Ψ{\rm Ψ}Ψ, namely K\mathcal{K}K-truncation and Y\mathcal{Y}Y-truncation, to obtain a kind of separated representation of Ψ{\rm Ψ}Ψ. Complexity analysis and several challenging experiments, including harmonic oscillators, Gaussian barrier scattering, electron-electron scattering, and a Helium-like system, all of which satisfy the separable assumption, confirm that the proposed DLRA algorithm has significant advantages, achieving a reduction in computational effort by one to two orders of magnitude in both runtime and memory requirements compared to the full-grid approach. It is worth noting that, even in the absence of a predetermined low-rank structure for the solution, DLRA can still serve as a numerical scheme that balances efficiency and accuracy.