Quantum Prediction of Transport Dynamics in Discretized State Spaces

2026-04-27

AI Summary

This work addresses the challenge of efficiently solving high-dimensional Bayesian state estimation, where the evolution of probability densities is hindered by prohibitive computational complexity. By leveraging the Fokker–Planck equation, the proposed method encodes probability densities into the amplitudes of quantum states within a discrete position-velocity space and implements the prediction step in the spectral domain via quantum Fourier transforms and phase rotations. The key innovation lies in the first exact unitary implementation of the drift term in amplitude space, complemented by a Wick-rotation-based unitary surrogate model for the diffusion term, thereby establishing a fully unitary propagation framework that overcomes the longstanding barrier of representing nonlinear diffusion in quantum amplitudes. Numerical experiments demonstrate excellent agreement with exact solutions of the Fokker–Planck equation and reveal exponential scalability in state dimensionality, substantially outperforming classical tensor decomposition approaches.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We propose a gate-based quantum algorithm for the prediction step of Bayesian state estimation based on the Fokker-Planck equation on a discretized position-velocity state space. The probability density is encoded in the amplitudes of a quantum state, enabling a compact representation of high-dimensional distributions. Exploiting the circulant structure of finite-difference operators, the evolution is realized in the spectral domain using quantum Fourier transforms and phase rotations. A key result is that the drift component can be implemented exactly in amplitude space, leading to an accurate reproduction of the classical transport dynamics. In contrast, the diffusion term does not admit a linear representation in amplitude space due to the nonlinear relation between probability density and wave function. To enable a quantum implementation, we introduce a unitary surrogate based on a Wick rotation, transforming diffusion into a dispersive phase evolution. This yields a fully unitary propagation that can be implemented efficiently on a gate-based quantum computer. The proposed method is evaluated numerically for different scenarios and shows strong agreement with the exact solution of the Fokker-Planck equation. The approach demonstrates the potential of quantum computing for Bayesian state estimation, as the representable state space grows exponentially with the number of qubits. This allows the efficient representation and propagation of probability densities that would otherwise require complex tensor decompositions on classical hardware, making the method a promising candidate for high-dimensional filtering problems.

Problem

• quantum prediction
• transport dynamics
• Fokker-Planck equation
• Bayesian state estimation
• discretized state space

Innovation

• quantum algorithm
• Fokker-Planck equation
• Bayesian state estimation
• quantum Fourier transform
• Wick rotation