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

TL;DR

This paper introduces a quantum algorithm that efficiently performs the diffusion step in grid-based Bayesian filters by leveraging quantum Fourier transforms, reducing quantum gate count and circuit depth compared to classical and other quantum methods.

Contribution

It presents a novel quantum approach for the diffusion step in Bayesian filtering that simplifies implementation and improves efficiency over existing methods.

Findings

Reproduces desired probability densities accurately

Requires fewer quantum gates than classical methods

Achieves shallower circuit depth in simulations

Abstract

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.