A Quantum Algorithm for the Diffusion Step of Grid-based Filter

We propose a simple quantum algorithm for implementing the diffusion step of grid-based Bayesian filters. The method encodes the advected state density and the process noise density into quantum registers and realizes diffusion using a quantum Fourier transform--based adder. This avoids the explicit convolution required in classical implementations and the repeated coin-flip operations used in quantum random walk approaches. Numerical simulations using a gate-based quantum computing simulator confirm that the proposed approach reproduces the desired probability densities while requiring significantly fewer quantum gates and much shallower circuit depth.

Lagrangian Grid-based Estimation of Nonlinear Systems with Invertible Dynamics

This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF) recently-developed for linear systems and extends it for systems with nonlinear dynamics that are invertible. The proposed nonlinear Lagrangian GbF reduces the computational complexity of the standard GbFs from quadratic to log-linear, while preserving all the strengths of the original GbF such as robustness, accuracy, and deterministic behaviour. The proposed filter is compared with the particle filter in several numerical studies using the publicly available MATLAB\textregistered\ implementation\footnote{https://github.com/pesslovany/Matlab-LagrangianPMF}.

Efficient Point Mass Predictor for Continuous and Discrete Models with Linear Dynamics

This paper deals with state estimation of stochastic models with linear state dynamics, continuous or discrete in time. The emphasis is laid on a numerical solution to the state prediction by the time-update step of the grid-point-based point-mass filter (PMF), which is the most computationally demanding part of the PMF algorithm. A novel way of manipulating the grid, leading to the time-update in form of a convolution, is proposed. This reduces the PMF time complexity from quadratic to log-linear with respect to the number of grid points. Furthermore, the number of unique transition probability values is greatly reduced causing a significant reduction of the data storage needed. The proposed PMF prediction step is verified in a numerical study.