Regression of Functions by Quantum Neural Networks Circuits

TL;DR

This work tackles automated quantum regression by synthesizing Reduced Regressor Quantum Neural Networks (RRQNNs) via a genetic algorithm, balancing circuit depth, gate configurations, and data re-uploading. It introduces regression-specific complexity metrics as meta-features to guide architecture selection, demonstrating high predictive accuracy in five scenarios and competitive performance against 17 classical baselines across 22 nonlinear functions. Although many classical models achieve strong mean performance, RRQNNs offer substantial parameter efficiency, and meta-learning enables data-driven model choice, suggesting a principled route toward hardware-aware quantum regression. The findings motivate further exploration of larger-scale, noisy-qubit settings and tighter integration with classical metalearning pipelines to realize practical quantum regression advantages.

Abstract

The performance of quantum neural network models depends strongly on architectural decisions, including circuit depth, placement of parametrized operations, and data-encoding strategies. Selecting an effective architecture is challenging and closely related to the classical difficulty of choosing suitable neural-network topologies, which is computationally hard. This work investigates automated quantum-circuit construction for regression tasks and introduces a genetic-algorithm framework that discovers Reduced Regressor QNN architectures. The approach explores depth, parametrized gate configurations, and flexible data re-uploading patterns, formulating the construction of quantum regressors as an optimization process. The discovered circuits are evaluated against seventeen classical regression models on twenty-two nonlinear benchmark functions and four analytical functions. Although classical methods often achieve comparable results, they typically require far more parameters, whereas the evolved quantum models remain compact while providing competitive performance. We further analyze dataset complexity using twelve structural descriptors and show, across five increasingly challenging meta-learning scenarios, that these measures can reliably predict which quantum architecture will perform best. The results demonstrate perfect or near-perfect predictive accuracy in several scenarios, indicating that complexity metrics offer powerful and compact representations of dataset structure and can effectively guide automated model selection. Overall, this study provides a principled basis for meta-learning-driven quantum architecture design and advances the understanding of how quantum models behave in regression settings--a topic that has received limited exploration in prior work. These findings pave the way for more systematic and theoretically grounded approaches to quantum regression.

Figures (18)

• Figure 1: An example of quantum circuit with one $\mathbf{CNOT}$ operator and two rotation gates, $\mathbf{Rx}$ and $\mathbf{Rz}$, one of them being controlled (controlled rotation).
• Figure 2: Traditional generic architecture of a parametric quantum circuit where the Parametric Layer can be repeated $n$ times.
• Figure 3: Learning process of a parametric quantum circuit. The classical optimizer iteratively updates the parameters $\boldsymbol{\theta}$ based on the feedback from quantum measurements.
• Figure 4: Strongly Entangling Layers (PennyLane implementation). Each layer applies three-parameter rotations $R(\alpha,\beta,\gamma)$ on every qubit, followed by CNOTs with periodic pattern: in layer $l$ with range $r=l \bmod M$, each qubit $i$ controls qubit $(i+r) \bmod M$. The weight tensor has shape $(L, M, 3)$bergholm2022pennylane.
• Figure 5: Basic Entangler Layers (PennyLane implementation). Each layer applies single-parameter rotations (default $R_X$) on all qubits, followed by a closed ring of CNOT gates connecting neighboring qubits with periodic boundary conditions. The weight tensor has shape $(L, M)$ where $L$ is the number of layers and $M$ the number of qubits; for two qubits, the periodic connection is omitted to avoid repeated entanglement bergholm2022pennylane.