We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.
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8 pages, 5 figues, code at https://github.com/wangleiphy/BohmianFlow
We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.
Bohmian mechanics is an alternative formulation of quantum mechanics that incorporates the familiar and intuitive picture of particles moving along trajectories and yet predicts the same results as the more widely accepted Copenhagen interpretation. In recent years there has been renewed interest in this Bohmian view, in part for the novel approach that it suggests to certain problems, such as decay processes, both from a theoretical and computational stand point. In this thesis we focus on using the concepts introduced by the Bohmian framework as a practical computational tool. I evaluate a number of implementations of the Bohmian method, get a sense of their strengths and weaknesses and attempt to overcome some stability issues that arise. For problems in one-dimension (lD), accurate solutions of the time-dependent Schrodinger equation produce a wave function from which Bohmian trajectories can be computed by integrating along flux lines. For direct integration of the quantum Hamilton-Jacobi equations, the main problems that arise are related to evauating the quantum potential (QP), especially in regions of low probability density. Sufficient accuracy is required to avoid unphysical trajectory crossings. A number of interpolation schemes were investigated, and smoothed splines with special treatment of edge effects gave the best results. For problems in 2D the alternating direction implicit (ADI) method was employed to produce the wave function. Ways of dealing with unphysical reflections from the boundaries of a finite size domain were studied. The use of cellular automata, especially the lattice-Boltzmann method (LBM) were also considered. Here Bohm trajectories would be propagated by following a small set of rules. The main problem identified is that, unless a scheme can be found in which the quantum potential is self-generating from an equation of continuity, the overhead of computing the QP at each time step, is prohibitive.
Studying the dynamics of open quantum systems can enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Since the density matrix ρ, which is the fundamental description for the dynamics of such systems, is high-dimensional, customized deep generative neural networks have been instrumental in modeling ρ. However, the complex-valued nature and normalization constraints of ρ, as well as its complicated dynamics, prohibit a seamless connection between open quantum systems and the recent advances in deep generative modeling. Here we lift that limitation by utilizing a reformulation of open quantum system dynamics to a partial differential equation (PDE) for a corresponding probability distribution Q, the Husimi Q function. Thus, we model the Q function seamlessly with off-the-shelf deep generative models such as normalizing flows. Additionally, we develop novel methods for learning normalizing flow evolution governed by high-dimensional PDEs based on the Euler method and the application of the time-dependent variational principle. We name the resulting approach Q-Flow and demonstrate the scalability and efficiency of Q-Flow on open quantum system simulations, including the dissipative harmonic oscillator and the dissipative bosonic model. Q-Flow is superior to conventional PDE solvers and state-of-the-art physics-informed neural network solvers, especially in high-dimensional systems.
The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.
Physics-Informed Neural Networks (PINNs) offer an effective framework for solving partial differential equations by incorporating governing physical laws into the learning process. When applied to quantum mechanical differential equations such as the Schr¨odinger equation, this approach produces smooth wavefunctions suitable for computing both conventional observables and Bohmian quantities. In this work, a PINNs based framework is developed to solve the one-dimensional time-dependent Schr¨odinger equation for the Quantum Har monic Oscillator, a fundamental model with broad relevance in quantum and plasma physics. A fully connected neural network is used to represent the real and imaginary parts of the wavefunction, with the Schr¨odinger equation enforced through the loss function together with appropriate initial and boundary conditions. Automatic differentiation is employed to compute the required derivatives and to extract physical quantities from the learned solution. The method accurately reproduces the ground-state wavefunction and yields a ground-state energy in agreement with the analytical result. The Bohm quantum potential obtained from the network also matches the corresponding analytical expression, demon strating that the essential spatial structure of the quantum amplitude is well captured. These results demonstrate that PINNs constitute a reliable and conceptually transparent framework for solving the time-dependent Schr¨odinger equation. Within this differential equation based setting, PINNs also enable systematic exploration of Bohmian aspects of quantum systems, where Bohmian quantities such as the quantum potential provide physical insight into confinement effects and the emergence of effective potentials.
In continuously monitored quantum systems, the feedback protocol of García-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian Hmeas=rA/τ applied with gain X tilts the distribution of measurement trajectories, with X<−2 producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained.
We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fréchet differentiation on the Banach space of trace-class operators, and Kähler geometry on the pure-state projective manifold, we prove that δlogPF/δρ=rA/τ=Hmeas. The García-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution -- exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators.
Two consequences follow. First, the feedback gain X generates a continuous one-parameter family of path measures (for feedback-active Hamiltonians with [H,A]=0), with X=−2 recovering the backward process in leading-order linearization -- a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods -- denoising score matching, sliced score matching -- to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments.
Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.