# QEKI: A Quantum–Classical Framework for Efficient Bayesian Inversion of PDEs

## TL;DR

This paper introduces QEKI, a quantum-classical framework that improves Bayesian inversion of PDEs by combining quantum neural networks with classical optimization.

## Contribution

The novel framework QEKI uses quantum neural networks with classical EKI to enable efficient Bayesian inversion with reduced parameter costs.

## Key findings

- QEKI achieves precise inversions of 1D and 2D nonlinear PDEs with noise.
- Quantum neural networks enable significant parameter compression compared to classical networks.
- The framework mitigates barren plateaus in quantum optimization via gradient-free EKI.

## Abstract

Solving Bayesian inverse problems efficiently stands as a major bottleneck in scientific computing. Although Bayesian Physics-Informed Neural Networks (B-PINNs) have introduced a robust way to quantify uncertainty, the high-dimensional parameter spaces inherent in deep learning often lead to prohibitive sampling costs. Addressing this, our work introduces Quantum-Encodable Bayesian PINNs trained via Classical Ensemble Kalman Inversion (QEKI), a framework that pairs Quantum Neural Networks (QNNs) with Ensemble Kalman Inversion (EKI). The core advantage lies in the QNN’s ability to act as a compact surrogate for PDE solutions, capturing complex physics with significantly fewer parameters than classical networks. By adopting the gradient-free EKI for training, we mitigate the barren plateau issue that plagues quantum optimization. Through several benchmarks on 1D and 2D nonlinear PDEs, we show that QEKI yields precise inversions and substantial parameter compression, even in the presence of noise. While large-scale applications are constrained by current quantum hardware, this research outlines a viable hybrid framework for including quantum features within Bayesian uncertainty quantification.