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

Yeongkwon Choe, Chan Gook Park, Jindřich Duník, Jan Krejčí, Jakub Matoušek, Marek Brandner

TL;DR

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.

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.

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

TL;DR

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.

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.
Paper Structure (14 sections, 27 equations, 2 figures, 1 table)

This paper contains 14 sections, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bayesian Estimation using Grid-based Filter
  3. Quantum State Estimation Algorithms: An Overview
  4. Goal of the Paper: Quantum Realized Diffusion GbF
  5. Preliminaries
  6. State-Space Model
  7. Grid-based Filtering and Problem Statement
  8. QFT-based Simple Quantum Diffusion
  9. Qubits and Their Representation
  10. Quantum Encoding of State and Noise Grids.
  11. Quantum Realization of the Diffusion Step
  12. Consistency Check: Two's-Complement Addition
  13. Numerical Implementation
  14. Conclusion

Figures (2)

  • Figure 1: Quantum circuit for the proposed QFT-based diffusion.
  • Figure 2: Approximate probability densities obtained via quantum sampling for the four test cases, with solid curves indicating the exact densities. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4