# Mitigating Exponential Mixed Frequency Growth through Frequency Selection

arXiv: https://arxiv.org/abs/2508.10533 · HTML: https://mlsanity.com/paper/2508.10533/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Let Quantum Neural Networks Choose Their Own Frequencies](https://mlsanity.com/paper/2309.03279/) (2023)
- [An exponentially-growing family of universal quantum circuits](https://mlsanity.com/paper/2212.00736/) (2022)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [Spectral Bias in Variational Quantum Machine Learning](https://mlsanity.com/paper/2506.22555/) (2025)
- [Fourier Analysis of Variational Quantum Circuits for Supervised Learning](https://mlsanity.com/paper/2411.03450/) (2024)
- [Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks](https://mlsanity.com/paper/2402.14515/) (2024)

## Built on by (most cited first, 1 of 1)

- [Long Range Frequency Tuning for QML](https://mlsanity.com/paper/2602.23409/) (2026)

## Abstract (arXiv)

Angle encoding has emerged as a popular feature map for embedding classical data into quantum models, naturally generating truncated Fourier series with universal function approximation capabilities. Despite this expressive capability, practical training faces significant challenges. Through controlled experiments with white-box target functions, we demonstrate that training failures can occur even when all established parameter sufficiency conditions are satisfied. Building on the redundancy-gradient framework of Duffy and Jastrzebski, we provide systematic experimental evidence that non-unique frequencies dominate the gradient landscape and crowd out target frequencies -- a burden that grows exponentially with encoding depth under unary encoding. Small-angle initialization mitigates this in one-dimensional settings but fails to scale to higher dimensions, where even ternary encoding -- which minimizes per-frequency redundancy -- faces intractable combinatorial growth of unique frequency tuples regardless of initialization or optimizer choice. We introduce frequency selection as a principled solution that restricts the model spectrum to only those frequencies present in the target function. For two-dimensional targets, frequency selection achieves near-optimal performance (median R2 ≈ 0.95) where dense approaches struggle, and remains tractable at high-frequency magnitudes where dense approaches fail entirely (median R2 ≈ 0.85). Validation on a real-world dataset confirms the approach transfers beyond synthetic settings.