  Many real-world problems, like modelling environment dynamics, physical
processes, time series etc., involve solving Partial Differential Equations
(PDEs) parameterised 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.
However, it results in a time complexity linear in the number of evaluations of
the PDEs while testing. Given the advancements in quantum hardware and the
recent results in quantum machine learning methods, we exploit the running
efficiency offered by these and propose quantum algorithms inspired by the
classical FNO, which result in time complexity logarithmic in the number of
evaluations and are, therefore, expected to be substantially faster than their
classical counterpart. At their core, we use the unary encoding paradigm and
orthogonal quantum layers and introduce a circuit to perform quantum Fourier
transform in the unary basis. We propose three different quantum circuits to
perform a quantum FNO. The proposals differ in their depth and their similarity
to the classical FNO. We also benchmark our proposed algorithms on three PDE
families, namely Burgers' equation, Darcy's flow equation and the Navier-Stokes
equation. The results show that our quantum methods are comparable in
performance to the classical FNO. We also perform an analysis on small-scale
image classification tasks where our proposed algorithms are at par with the
performance of classical CNNs, proving their applicability to other domains as
well.
