Quantum Image Denoising: A Framework via Boltzmann Machines, QUBO, and Quantum Annealing

TL;DR

This paper presents a quantum-compatible framework for binary image denoising using Boltzmann machines and QUBO, demonstrating its effectiveness on quantum annealers and classical heuristics.

Contribution

It introduces a novel QUBO formulation for image denoising based on RBMs, with theoretical analysis and empirical validation on quantum hardware and classical methods.

Findings

Denoised images are closer to noise-free images in expectation.

The QUBO formulation is suitable for quantum annealing implementation.

The method is robust and applicable to any binary data.

Abstract

We investigate a framework for binary image denoising via restricted Boltzmann machines (RBMs) that introduces a denoising objective in quadratic unconstrained binary optimization (QUBO) form and is well-suited for quantum annealing. The denoising objective is attained by balancing the distribution learned by a trained RBM with a penalty term for derivations from the noisy image. We derive the statistically optimal choice of the penalty parameter assuming the target distribution has been well-approximated, and further suggest an empirically supported modification to make the method robust to that idealistic assumption. We also show under additional assumptions that the denoised images attained by our method are, in expectation, strictly closer to the noise-free images than the noisy images are. While we frame the model as an image denoising model, it can be applied to any binary data. As the QUBO formulation is well-suited for implementation on quantum annealers, we test the model on a D-Wave Advantage machine, and also test on data too large for current quantum annealers by approximating QUBO solutions through classical heuristics.