  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.
