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.

翻译：量子神经网络（QNN）是量子机器学习中的一种常见方法，因其与变分量子电路的紧密联系，成为在含噪中等规模量子（NISQ）设备上实际应用的有前景候选方案。QNN可表示为有限傅里叶级数，其中频率集合称为频谱。我们分析了该频谱，并针对一大类模型证明了多种极大性结果。此外，我们证明了在温和条件下，具有相同面积 $A = RL$ 的模型类之间存在保持频谱的双射，其中 $R$ 表示量子比特数，$L$ 表示层数，我们因此将其称为面积保持变换下的谱不变性。借此，我们解释了文献中常观察到的 $R$ 和 $L$ 的对称性，并表明最大频谱仅取决于面积 $A = RL$，而非 $R$ 和 $L$ 的个别值。进一步，我们扩展了现有结果，将具有任意层数的QNN的最大可能频谱明确表示为生成器频谱的函数。若QNN的生成器可进一步分解为二维子生成器，则该表示遵循基本数论考量。对于任意维度生成器的情况，我们基于所谓的哥隆尺扩展了现有结果，并引入了一种基于转盘问题变体的新方法，我们称之为松弛转盘问题。