Title: Quantum spectral operator learning for solving partial differential equations
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: Variational Quantum Linear Solver, Spectral Operator Learning, Operator Learning, Quantum Machine Learning, Spectral Methods

Abstract: Partial differential equations (PDEs) are central to modeling physical and engineering systems. Operator learning approximates their solution operators, enabling fast inference after training across diverse problem instances and strong generalization. While recent advances have proposed unsupervised methods that mitigate the cost of data generation, classical neural network–based approaches remain computationally expensive for high-dimensional operators and fine-resolution problems. To address these challenges, we propose a quantum--classical hybrid framework for unsupervised spectral operator learning. Our approach predicts spectral coefficients using quantum circuits, with gate parameters mapped from PDE instances (e.g., forcing functions or PDE parameters) via a classical neural network. To improve efficiency and feasibility, we introduce a training objective that requires fewer measurement repetitions than standard variational quantum linear solvers (VQLS). With this, we design shallower circuits by replacing controlled-unitary gates with direct Pauli measurements, which in turn allows grouping of commuting measurement operators for further reduction in runtime. The objective also resolves the sign ambiguity inherent in standard VQLS and guarantees recovery of the correct solution sign for PDEs. Overall, our framework reduces the computational cost and improves solution accuracy of VQLS, while also demonstrating the potential efficiency and scalability advantages of quantum operator learning over classical machine learning approaches. We validate our framework on one- and two-dimensional reaction--diffusion, Helmholtz, and convection--diffusion equations under diverse boundary conditions, achieving relative errors below $1\%$.

TL;DR: We propose a novel quantum--classical hybrid framework for spectral operator learning that reduces computational cost and efficiently solves diverse PDEs.