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.

翻译：针对基于福克-普朗克方程在离散化位置-速度状态空间中的贝叶斯状态估计预测步骤，我们提出了一种基于量子门的量子算法。概率密度被编码于量子态振幅中，从而实现对高维分布的紧凑表示。通过利用有限差分算子的循环结构，可在频谱域中借助量子傅里叶变换与相位旋转实现概率演化。核心成果在于漂移分量可在振幅空间中精确实现，从而准确复现经典输运动力学。与之相对，扩散项因概率密度与波函数间存在非线性关系，无法在振幅空间中获得线性表示。为促成量子实现，我们引入基于威克变换的酉替代方案，将扩散转化为色散相位演化，进而构建可高效运行于量子门计算机的全酉传播过程。通过不同场景下的数值评估，所提方法与福克-普朗克方程的精确解高度吻合。该研究展示了量子计算在贝叶斯状态估计中的潜力：可表示的状态空间随量子比特数指数增长，能高效表示与传播在经典硬件上需复杂张量分解的概率密度，使其成为高维滤波问题的理想候选方案。