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 $A:=R\cdot L$ 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.

翻译：我们利用闵可夫斯基和分析了量子神经网络（QNNs）的频率谱，这给出了一个紧凑的代数描述并允许显式计算。基于此描述，我们证明了针对广泛QNN架构类别的若干极大性结果。在一些温和的技术条件下，我们建立了具有相同面积 $A:=R\cdot L$ 的模型类别之间的双射，该双射保持频率谱不变，其中 $R$ 表示量子比特数，$L$ 表示层数，我们因此称之为面积保持变换下的谱不变性。由此，我们解释了文献中经常观察到的关于 $R$ 和 $L$ 的结果对称性，并证明了最大频率谱仅取决于面积 $A=RL$，而不依赖于 $R$ 和 $L$ 各自的取值。此外，我们收集并扩展了现有结果，并具体给出了具有任意层数的QNN的最大可能频率谱，该频率谱是其生成元谱的函数。在任意维生成元的情况下，我们引入的两种极大性概念有所不同，我们扩展了现有的基于Golomb尺的结果，并引入了一种基于变体turnpike问题（我们称之为松弛turnpike问题）的第二种新方法。我们全面阐明了对于给定面积 $A$，应如何选择QNN的生成元才能获得最大的频率谱，从而为更深层的理论理解做出贡献。然而，我们的数值实验表明，可训练性不仅取决于 $A = RL$，还取决于 $(R,L)$ 的选择，因此仅了解最大频率谱并不足以确保良好的可训练性。