A Quantum Algorithm for the Prediction Step of a Bayesian Recursion

The prediction step is a crucial element of the Bayesian recursion for target tracking and state estimation in general. Discrete representations of the probability density function (pdf) can deal with non-linear models and nonGaussian noise, however, the prediction step is challenging to solve on classical computers. In this paper, a novel concept of quantum simulation to solve the application of a continuous noise motion model to a pdf is presented. The pdf is prepared as the squared amplitudes for the basis states spanned by the number of used qubits. The number of required qubits grows linearly in the number of time steps to simulate in a prediction phase. One step performs a single Brownian motion, which is used to generate the diffusion of the Wiener increments. A drift function can be implemented based on a separate register, which holds the pdf on the velocity information for an adequate discretization. The approach will be visualized in terms of quantum circuits and evaluated based on a quantum simulator.