Benchmarking Encoding Families in Quantum Neural Networks Under Fixed Circuit Area for Frequency Spectrum and Trainability

Abstract

Quantum Neural Networks (QNNs) offer a promising framework for integrating quantum computing principles into machine learning, yet their practical capabilities and limitations remain insufficiently studied. In this work, we systematically investigate the trainability and approximation properties of QNNs by benchmarking diverse circuit architectures and encoding strategies across synthetic and real-world datasets. We analyze several ansätze, including Hamming, binary, exponential, ternary, turnpike and Golomb, by evaluating their ability to learn synthetic data modeled as random finite Fourier series. To assess real-world applicability, we further evaluate QNNs on two time-series classification tasks: a Fischertechnik pneumatic leak detection dataset and the publicly available NASA bearing fault dataset. Our experiments show that while broader frequency spectra can theoretically enhance expressivity, practical trainability is strongly influenced by architectural factors such as qubit count and circuit depth. Notably, we find that QNNs perform best when the frequency spectrum is tailored to the target function's complexity but remains as compact as possible. Moreover, architectures with identical frequency spectra can differ in trainability, with configurations using more qubits and fewer layers generally performing better, except in the single-layer case. These findings provide guidelines for selecting QNN ansätze and offer new insights into the interplay between expressivity and trainability in quantum machine learning.

Figures (7)

• Figure 1: Generic strongly entangling model circuit architecture for 6 qubits. The circuit consists of two trainable block layers$B_1$ and $B_2$ with a range of controls of $r=1$ and $r=2$ respectively. The circuit consists of 12 trainable single-qubit gates $G = G(\alpha, \beta, \gamma)$ and 12 controlled single-qubit gates.
• Figure 2: The learning capability defined as the mean of all runs together with the 50% percentile of all 100 runs with $g \in G_K$.
• Figure 3: Time-series of the Fischertechnik dataset. Left: measured current. Right: corresponding power. The vertical dashed line indicates the onset of a pneumatic leak. Active (nonzero) and inactive (zero) operating periods are visible in both signals.
• Figure 4: Training loss versus epoch for four encoding strategies (Hamming, binary, exponential, and ternary) with $R=2$ and $L=3$ ($A=6$), trained for 3000 epochs at learning rate $\eta=0.005$ (other hyperparameters fixed). Inset: zoom of the late-epoch region to highlight differences in the converged loss values and transient behavior.
• Figure 5: A simplified depiction of a ball bearing. For the second NASA dataset, the outer race of bearing 1 failed.