Fourier Analysis of Variational Quantum Circuits for Supervised Learning

TL;DR

This paper applies Fourier analysis to variational quantum circuits, revealing how variational parameters influence the spectrum and enabling prediction of the circuit's data fitting capabilities.

Contribution

It provides the first detailed description of Fourier coefficients as trigonometric polynomials and an algorithm to compute the exact spectrum of VQCs.

Findings

The spectrum depends on variational parameters constraining Fourier coefficients.

An algorithm to compute the exact Fourier spectrum of VQCs.

Ability to predict the best VQC for data fitting based on Fourier analysis.

Abstract

VQC can be understood through the lens of Fourier analysis. It is already well-known that the function space represented by any circuit architecture can be described through a truncated Fourier sum. We show that the spectrum available to that truncated Fourier sum is not entirely determined by the encoding gates of the circuit, since the variational part of the circuit can constrain certain coefficients to zero, effectively removing that frequency from the spectrum. To the best of our knowledge, we give the first description of the functional dependence of the Fourier coefficients on the variational parameters as trigonometric polynomials. This allows us to provide an algorithm which computes the exact spectrum of any given circuit and the corresponding Fourier coefficients. Finally, we demonstrate that by comparing the Fourier transform of the dataset to the available spectra, it is possible to predict which VQC out of a given list of choices will be able to best fit the data.