Regression and Classification with Single-Qubit Quantum Neural Networks

TL;DR

The paper proposes the Single-Qubit Quantum Neural Network (SQQNN), a resource-efficient quantum neural architecture built from parameterized single-qubit rotations for both regression and binary classification on near-term quantum hardware. It introduces gradient-descent training for regression and a polynomial-based linear least squares method for fast classification, enabling efficient optimization with minimal qubits. Through extensive experiments on logic gates, the sinc function, and real-world regression tasks, as well as on synthetic and real datasets for classification (including MNIST), the SQQNN demonstrates high accuracy, robustness to noise, and scalable performance as the network depth (number of neurons) increases. The work highlights the practicality of minimal-qubit quantum networks for contemporary quantum devices and outlines future directions toward multi-qubit systems, novel activations, and qudit-based extensions.

Abstract

Since classical machine learning has become a powerful tool for developing data-driven algorithms, quantum machine learning is expected to similarly impact the development of quantum algorithms. The literature reflects a mutually beneficial relationship between machine learning and quantum computing, where progress in one field frequently drives improvements in the other. Motivated by the fertile connection between machine learning and quantum computing enabled by parameterized quantum circuits, we use a resource-efficient and scalable Single-Qubit Quantum Neural Network (SQQNN) for both regression and classification tasks. The SQQNN leverages parameterized single-qubit unitary operators and quantum measurements to achieve efficient learning. To train the model, we use gradient descent for regression tasks. For classification, we introduce a novel training method inspired by the Taylor series, which can efficiently find a global minimum in a single step. This approach significantly accelerates training compared to iterative methods. Evaluated across various applications, the SQQNN exhibits virtually error-free and strong performance in regression and classification tasks, including the MNIST dataset. These results demonstrate the versatility, scalability, and suitability of the SQQNN for deployment on near-term quantum devices.

Figures (7)

• Figure 1: Representation of the most general quantum neuron with an arbitrary input and measurement. The output corresponds to the average value obtained from measuring the observable $\mathcal{O}$.
• Figure 2: Illustration of the reduced quantum neuron with a fixed input state and measurement in the computational basis.
• Figure 3: Representation of the SQQNN with an arbitrary input and measurement. The output is the average value obtained from measuring the observable $\mathcal{O}$. Each neuron $\mathcal{N}_k$ has three trainable parameters, while the input state ${\left\vert{\psi}\right\rangle}$ and the observable $\mathcal{O}$ each have a single trainable parameter, since we have set $\phi=\varphi=0$.
• Figure 4: Representation of the reduced SQQNN. Each neuron $\mathcal{N}_k$ is the rotation operator $R_y(\beta_k)$, which has only one trainable parameter.
• Figure 5: Performance of the SQQNN on a sinc function regression task using a single neuron. The top-left panel shows the noiseless training and test data combined, while the bottom-left panel displays the results with added white noise. The top-right and bottom-right panels depict the corresponding error as a function of the number of iterations.