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：量子神经网络是量子机器学习中广泛采用的方法，其与变分量子线路的紧密联系使其成为含噪中等规模量子设备上具有实际应用前景的候选方案。量子神经网络可表示为有限傅里叶级数，其中频率集合被称为频率谱。本文分析该频率谱，针对一大类模型证明了多种极大性结果。此外，我们证明在温和条件下，具有相同面积$A=RL$的模型类之间存在保持频率谱的双射（其中$R$表示量子比特数，$L$表示层数），因而称之为面积保持变换下的谱不变性。基于此，我们解释了文献中普遍观测到的$R$与$L$对称性，并证明最大频率谱仅取决于面积$A=RL$，而非$R$和$L$的独立取值。进一步，我们将现有结果推广，以生成元谱的函数形式刻画了具有任意层数量子神经网络的最大可能频率谱。若量子神经网络生成元可进一步分解为二维子生成元，则该刻画可通过初等数论分析直接获得。对于任意维数生成元的情形，我们基于所谓Golomb尺子扩展了现有结果，并提出了基于转向问题变体的第二种新颖方法——松弛转向问题。