# Fourier Analysis of Variational Quantum Circuits for Supervised Learning

arXiv: https://arxiv.org/abs/2411.03450 · HTML: https://mlsanity.com/paper/2411.03450/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 5 Nov 2024.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Parametrized quantum policies for reinforcement learning](https://mlsanity.com/paper/2103.05577/) (2021)
- [A Survey on Quantum Reinforcement Learning](https://mlsanity.com/paper/2211.03464/) (2022)
- [Quantum reinforcement learning in continuous action space](https://mlsanity.com/paper/2012.10711/) (2020)
- [Let Quantum Neural Networks Choose Their Own Frequencies](https://mlsanity.com/paper/2309.03279/) (2023)
- [Quantum Policy Gradient Algorithm with Optimized Action Decoding](https://mlsanity.com/paper/2212.06663/) (2022)
- [Quantum Reinforcement Learning via Policy Iteration](https://mlsanity.com/paper/2203.01889/) (2022)
- [High Dimensional Quantum Machine Learning With Small Quantum Computers](https://mlsanity.com/paper/2203.13739/) (2022)
- [An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-based Gradients](https://mlsanity.com/paper/2305.00224/) (2023)
- [Incremental Data-Uploading for Full-Quantum Classification](https://mlsanity.com/paper/2205.03057/) (2022)
- [Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule](https://mlsanity.com/paper/2404.15751/) (2024)

## Built on by (most cited first, 4 of 4)

- [Spectral Bias in Variational Quantum Machine Learning](https://mlsanity.com/paper/2506.22555/) (2025)
- [Mitigating Exponential Mixed Frequency Growth through Frequency Selection](https://mlsanity.com/paper/2508.10533/) (2025)
- [Long Range Frequency Tuning for QML](https://mlsanity.com/paper/2602.23409/) (2026)
- [Watching Quantum Models Think: Hilbert-Space Interpretability in Quantum Transformer Blocks](https://mlsanity.com/paper/2609.23016/) (2026)

## Abstract (arXiv)

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.