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

• URL: http://arxiv.org/abs/2402.14515v2
• Date: Mon, 11 Mar 2024 15:40:18 GMT
• Title: Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks
• Authors: Patrick Holzer, Ivica Turkalj
• Abstract summary: We show that the maximal frequency spectrum depends only on the area $A = RL$ and not on the individual values of $R$ and $L$. We also specify the maximum possible frequency spectrum of a QNN with arbitrarily many layers as a function of the spectrum of its generators.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning due to their close connection to Variational Quantum Circuits, making them a promising candidate for practical applications on Noisy Intermediate-Scale Quantum (NISQ) devices. A QNN can be expressed as a finite Fourier series, where the set of frequencies is called the frequency spectrum. We analyse this frequency spectrum and prove, for a large class of models, various maximality results. Furthermore, we prove that under some mild conditions there exists a bijection between classes of models with the same area $A = RL$ 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 extend existing results and specify the maximum possible frequency spectrum of a QNN with arbitrarily many layers as a function of the spectrum of its generators. If the generators of the QNN can be further decomposed into 2-dimensional sub-generators, then this specification follows from elementary number-theoretical considerations. In the case of arbitrary dimensional generators, we extend existing results based on the so-called Golomb ruler and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem.

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