Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks
This provides theoretical insights for researchers in quantum machine learning, though it appears incremental as it extends existing results with new algebraic tools and notions.
The paper tackles the problem of analyzing the frequency spectrum of Quantum Neural Networks (QNNs) by proving spectral invariance under area-preserving transformations and maximality results, showing that the maximal frequency spectrum depends only on the area A=RL, not on individual qubits R and layers L.
Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning. We analyze this frequency spectrum using the Minkowski sum for sets and the set of differences, which makes it particularly easy to express and calculate the frequency spectrum algebraically, and prove different maximality results for a large class of models. Furthermore, we prove that under some mild conditions there exists 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 arbitrarily many layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduces notions of maximality differ, 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.