Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks

TL;DR

The paper investigates how the frequency spectrum of Quantum Neural Networks behaves under architectural transformations, showing that under area-preserving changes (keeping $A=RL$ fixed) the spectrum remains invariant. It develops a finite Fourier-series framework for QNNs and proves spectral invariance, implying the spectrum depends only on the area rather than the specific qubit/depth split. It then derives maximality results: for 2‑D sub-generators the spectrum is maximized in a precise, shape-dependent way, and for arbitrary-dimensional generators it uses Golomb rulers to maximize spectrum size and introduces a relaxed turnpike problem to maximize the largest contiguous spectral interval. The numerical results illustrate trade-offs between spectral richness and trainability, indicating that larger maximal spectra do not guarantees easier optimization. Collectively, the work provides a principled, algebraic understanding of QNN expressivity bounds and design principles for choosing generators and architectures within resource constraints.

Abstract

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.

Figures (7)

• Figure 1: Circuits of the parallel and the sequential ansatz. $R$ is the number of qubits per variable, $L$ the number of layers per variable $x_n$ and $N$ is the dimension of $\boldsymbol x \in \mathbb{R}^N$. Note that in total the parallel ansatz needs $N \cdot R$ many qubits and $L$ layers in total, while the model for the sequential ansatz has $R$ qubits and $N\cdot L$ layers in total. The parameter encoding layers are coloured red, the data encoding layers are coloured blue. The data encoding layers have the form $S(x) = e^{-ixH}$ for some Hamiltonian $H$ called generators. The generators are typically composed of smaller sub-generators. We have illustrated that all generators are composed of $2\times 2$ matrices and thus acting on a single qubit each. The sub-generators have been marked as white squares.
• Figure 2: Visualisation of Theorem \ref{['theorem: Spectral Invariance Under Area-Preserving Transformations']}. By Theorem \ref{['theorem: Univariate QNN is Fourier']}, the frequency spectrum only depends on the sub-generators $H_{r, l}$. The QNN can thus be represented by a rectangle containing these sub-generators as squares in the arrangement matching their occurrence in the quantum circuit. By Theorem \ref{['theorem: Spectral Invariance Under Area-Preserving Transformations']}, the arrangement is irrelevant for the frequency spectrum, as there exists an area-preserving transformation for all other compatible rectangles.
• Figure 3: Example of how to extend Theorem \ref{['theorem: Spectral Invariance Under Area-Preserving Transformations']} to QNNs without the requirement that all generators are $k$-dimensional. The sides of the individual rectangles representing a $k_{r, l}=2^{q_{r, l}}$-dimensional sub-generator $H_{r, l}$ have side lengths $q_{r, l} \times 1$. In this example, we used $q=1, 2, 3, 4$, $(R, L) = (6, 6)$ and $(R', L') = (4, 9)$. An area-preserving transformation is only possible if the target rectangular could be tiled with the given sub-generator rectangles.
• Figure 4: Frequency distributions for the considered QNN encoding schemes. The degeneracy indicates the number of distinct parameter combinations giving rise to the same frequency, i.e., the number of Fourier terms sharing that frequency. As in Table \ref{['tab: numerical examples frequency spectra']}, we use $H = G_8$ for the Golomb encoding and $H = T_8$ for the turnpike encoding; for all other encodings, we set $H = Z/2$. The circuit shape is $(2,2)$ for the Hamming, binary, ternary, and equal-layer encodings, and $(3,1)$ for the Golomb and turnpike encodings.
• Figure 5: Approximation of the target function $g(x) = \frac{1}{20}\sum_{n=1}^9 \sin(n x)$ using different encoding schemes. All models have circuit shape $(3,1)$. The sub-generator $H$ is chosen as $Z/2$ for the Hamming, binary, ternary, and equal-layer encodings, $G_8$ for the Golomb encoding, and $T_8$ for the turnpike encoding. The interval $[-\pi,\pi]$ is discretized into 1000 equidistant points, which are used as training targets. Model training is performed using ADAM with $b_1=0.9$, $b_2=0.999$, and $\epsilon = 10^{-8}$, treating the full dataset as a single batch per epoch. The encoding unitaries $W_{\boldsymbol{\theta}}^{(l)}$ are implemented using Pennylane’s StronglyEntanglingLayers with 10 entangling layers. All models are trained for 10,000 epochs with learning rate $10^{-4}$ and mean squared error loss.

Theorems & Definitions (35)

• Definition 1: Kronecker Sum
• Lemma 2: Properties of the Kronecker Sum
• Definition 3
• Lemma 4
• Definition 5: Parallel and Sequential Ansatz
• Definition 6: Quantum Neural Networks
• Theorem 7: Univariate QNN is a Fourier Series
• Theorem 8: Frequency Spectrum of a Multivariate QNN
• Theorem 9: Spectral Invariance Under Area-Preserving Transformations
• proof