QuanONet: Quantum Neural Operator with Application to Differential Equation

Differential equations are essential and popular in science and engineering. Learning-based methods including neural operators, have emerged as a promising paradigm. We explore its quantum counterpart, and propose QuanONet -- a quantum neural operator which has not been well studied in literature compared with their counterparts in other machine learning areas. We design a novel architecture as a hardware-efficient ansatz, in the era of noisy intermediate-scale quantum (NISQ). Its circuit is pure quantum. By lying its ground on the operator approximation theorem for its quantum counterpart, QuanONet in theory can fit various differential equation operators. We also propose its modified version TF-QuanONet with ability to adaptively fit the dominant frequency of the problem. The real-device empirical results on problems including anti-derivative operators, Diffusion-reaction Systems demonstrate that QuanONet outperforms peer quantum methods when their model sizes are set akin to QuanONet.

Differential equations are like mathematical recipes that describe how things change over time or space — from the weather and fluid flow to biological processes and engineering systems. Traditionally, solving these equations can be slow and complicated, especially for complex problems.Recently, new learning methods inspired by artificial intelligence have shown promise in speeding up and improving these solutions. Our work takes this idea a step further by exploring how quantum computers — powerful machines that use the strange rules of quantum physics — can help solve these equations even more efficiently.We introduce QuanONet, a new kind of quantum-based learning model designed specifically for this task. It’s built to work well with today’s early quantum devices, which are still limited but rapidly improving. Our approach allows QuanONet to learn and predict solutions to a wide variety of differential equations. We also developed an improved version called TF-QuanONet, which can automatically focus on the most important features of the problem for better accuracy.Testing our models showed that QuanONet outperforms other similar quantum approaches, demonstrating a promising step toward practical quantum-enhanced scientific computing.