# Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks

arXiv: https://arxiv.org/abs/2402.14515 · HTML: https://mlsanity.com/paper/2402.14515/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Builds on (most cited first, 10 of 10)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (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)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Input Redundancy for Parameterized Quantum Circuits](https://mlsanity.com/paper/1901.11434/) (2019)
- [An exponentially-growing family of universal quantum circuits](https://mlsanity.com/paper/2212.00736/) (2022)
- [Fock State-enhanced Expressivity of Quantum Machine Learning Models](https://mlsanity.com/paper/2107.05224/) (2021)

## Built on by (most cited first, 3 of 3)

- [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)
- [Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency](https://mlsanity.com/paper/2605.05942/) (2026)

## Abstract (arXiv)

We analyze the frequency spectrum of Quantum Neural Networks (QNNs) using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area A:=R· L that preserves the frequency spectrum, where R denotes the number of qubits and L the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in R and L in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area A=RL and not on the individual values of R and L. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area A, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on A = RL, but also on the choice of (R,L), so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability.