# Let Quantum Neural Networks Choose Their Own Frequencies

arXiv: https://arxiv.org/abs/2309.03279 · HTML: https://mlsanity.com/paper/2309.03279/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 53 references that are in the map.
- First posted on arXiv 6 Sep 2023.

## Builds on (most cited first, 23 of 23)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Transfer learning in hybrid classical-quantum neural networks](https://mlsanity.com/paper/1912.08278/) (2019)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Recurrent Quantum Neural Networks](https://mlsanity.com/paper/2006.14619/) (2020)
- [Training Quantum Embedding Kernels on Near-Term Quantum Computers](https://mlsanity.com/paper/2105.02276/) (2021)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Input Redundancy for Parameterized Quantum Circuits](https://mlsanity.com/paper/1901.11434/) (2019)
- [Quantum Self-Supervised Learning](https://mlsanity.com/paper/2103.14653/) (2021)
- [Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes](https://mlsanity.com/paper/2304.11247/) (2023)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [Quantum Kernel Alignment with Stochastic Gradient Descent](https://mlsanity.com/paper/2304.09899/) (2023)
- [A super-polynomial quantum-classical separation for density modelling](https://mlsanity.com/paper/2210.14936/) (2022)
- [Quantum Model-Discovery](https://mlsanity.com/paper/2111.06376/) (2021)
- [Single-component gradient rules for variational quantum algorithms](https://mlsanity.com/paper/2106.01388/) (2021)
- [Harmonic (Quantum) Neural Networks](https://mlsanity.com/paper/2212.07462/) (2022)

## Built on by (most cited first, 22 of 22)

- [Training robust and generalizable quantum models](https://mlsanity.com/paper/2311.11871/) (2023)
- [Quantum Neural Networks in Practice: A Comparative Study with Classical Models from Standard Data Sets to Industrial Images](https://mlsanity.com/paper/2411.19276/) (2024)
- [Spectral Bias in Variational Quantum Machine Learning](https://mlsanity.com/paper/2506.22555/) (2025)
- [QUACK: Quantum Aligned Centroid Kernel](https://mlsanity.com/paper/2405.00304/) (2024)
- [Fourier Analysis of Variational Quantum Circuits for Supervised Learning](https://mlsanity.com/paper/2411.03450/) (2024)
- [Spectral methods: crucial for machine learning, natural for quantum computers?](https://mlsanity.com/paper/2603.24654/) (2026)
- [Evaluating Angle and Amplitude Encoding Strategies for Variational Quantum Machine Learning: their impact on model's accuracy](https://mlsanity.com/paper/2508.00768/) (2025)
- [Expressivity of deterministic quantum computation with one qubit](https://mlsanity.com/paper/2411.02751/) (2024)
- [Quantum Graph Attention Networks: Trainable Quantum Encoders for Inductive Graph Learning](https://mlsanity.com/paper/2509.11390/) (2025)
- [Exponential capacity scaling of classical GANs compared to hybrid latent style-based quantum GANs](https://mlsanity.com/paper/2601.05036/) (2026)
- [Quantum Circuit Training with Growth-Based Architectures](https://mlsanity.com/paper/2411.16560/) (2024)
- [QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representations](https://mlsanity.com/paper/2504.19053/) (2025)
- [Mitigating Exponential Mixed Frequency Growth through Frequency Selection](https://mlsanity.com/paper/2508.10533/) (2025)
- [Can Geometric Quantum Machine Learning Lead to Advantage in Barcode Classification?](https://mlsanity.com/paper/2409.01496/) (2024)
- [The interplay of robustness and generalization in quantum machine learning](https://mlsanity.com/paper/2506.08455/) (2025)
- [CutReg: A loss regularizer for enhancing the scalability of QML via adaptive circuit cutting](https://mlsanity.com/paper/2506.14858/) (2025)
- [Re-uploading quantum data: a universal function approximator for quantum inputs](https://mlsanity.com/paper/2509.18530/) (2025)
- [Quantum Generator Kernels](https://mlsanity.com/paper/2602.00361/) (2026)
- [Long Range Frequency Tuning for QML](https://mlsanity.com/paper/2602.23409/) (2026)
- [Latent Style-based Quantum Wasserstein GAN for Drug Design](https://mlsanity.com/paper/2603.22399/) (2026)
- [Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations](https://mlsanity.com/paper/2608.27626/) (2026)
- [Fourier Analysis of Parametrized Interactive Quantum Classifiers](https://mlsanity.com/paper/2609.17991/) (2026)

## Abstract (arXiv)

Parameterized quantum circuits as machine learning models are typically well described by their representation as a partial Fourier series of the input features, with frequencies uniquely determined by the feature map's generator Hamiltonians. Ordinarily, these data-encoding generators are chosen in advance, fixing the space of functions that can be represented. In this work we consider a generalization of quantum models to include a set of trainable parameters in the generator, leading to a trainable frequency (TF) quantum model. We numerically demonstrate how TF models can learn generators with desirable properties for solving the task at hand, including non-regularly spaced frequencies in their spectra and flexible spectral richness. Finally, we showcase the real-world effectiveness of our approach, demonstrating an improved accuracy in solving the Navier-Stokes equations using a TF model with only a single parameter added to each encoding operation. Since TF models encompass conventional fixed frequency models, they may offer a sensible default choice for variational quantum machine learning.