A Fourier Lens on Parameterised Quantum Circuits: A Review

Variational quantum algorithms (VQAs) are among the most promising near-term approaches for quantum machine learning. For architectures that encode classical data and trainable angles via unitary evolutions generated by Hermitian operators, the expectation values of parameterised quantum circuits (PQCs) can be written as finite Fourier series in input variables. In this spectral view, the set of candidate frequencies is fixed by the eigenvalue structure of the data-encoding Hamiltonians, while the extent to which this spectrum is effectively usable depends on how flexibly the associated Fourier coefficients can be tuned through both the encoding and trainable circuit blocks; we review how these design choices determine spectral richness, degeneracy, coefficient correlations, and their trade-offs with trainability via a Fourier-based account of barren plateaus. We further discuss generalisation and function-approximation performance, including conditions under which overparameterised PQCs can interpolate noisy data yet maintain low test error and when angle-encoding with data re-uploading can realise universal trigonometric approximators for periodic functions. We conclude by highlighting open problems in architecture design, scalable coefficient control, and need for analysis on generalisation capabilities.