Quantum Fourier Networks for solving Parametric PDEs

Nishant Jain, Jonas Landman, Natansh Mathur, Iordanis Kerenidis

Keywords: quantum computing, quantum machine learning, quantum deep learnin, fourier transform, fourier neural operator, PDE, partial differential equation

TL;DR: We provide three new quantum circuits to reproduce the Fourier Neural Operator, in order to learn PDEs solutions, and tested them on practical use cases.

Abstract: Many real-world problems like modelling environment dynamics, physical processes, time series etc., involve solving Partial Differential Equations (PDEs) parameterized by problem-specific conditions. Recently, a deep learning architecture called Fourier Neural Operator (FNO) proved to be capable of learning solutions of given PDE families, for any initial conditions as input. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we propose three quantum circuits, inspired by the FNO, to learn this functional mapping for PDEs. The proposed algorithms are distinguished based on the trade-off between depth and their similarity to the classical FNO. At their core, we make use of unary encoding paradigm and orthogonal quantum layers, and introduce a new quantum Fourier transform in the unary basis. With respect to the number of samples, our quantum algorithm is proven to be substantially faster than the classical counterpart. We benchmark our proposed algorithms on three PDE families, namely Burger’s equation, Darcy’s flow equation and the Navier-Stokes equation, and the results show that our quantum methods are comparable in performance to the classical FNO. We also show an analysis of the image classification tasks where our proposed algorithms are able to match the accuracy of the CNNs, thereby showing their applicability to other domains.

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