Regression and Classification with Single-Qubit Quantum Neural Networks

Technical Summary: Regression and Classification with Single-Qubit Quantum Neural Networks

Problem Statement
The paper addresses the need for resource-efficient and scalable Quantum Neural Networks (QNNs) suitable for near-term quantum devices (NISQ). While classical machine learning has transformed pattern recognition and decision-making, quantum computing offers potential advantages through parallelism and entanglement. However, many existing QNN architectures require complex multi-qubit circuits that are prone to noise and decoherence. The authors propose a Single-Qubit Quantum Neural Network (SQQNN) to perform both regression and binary classification tasks using minimal quantum resources, specifically leveraging single-qubit neurons and parameterized unitary operators.

Methodology
The proposed SQQNN architecture is built upon a generalized single-qubit quantum neuron model.

• Neuron Model: The fundamental unit is a single qubit subjected to a sequence of rotation gates $R_z(\alpha)R_y(\beta)R_z(\gamma)$, where the angles are functions of the input data. The output is the expectation value of an observable $O$, measured after a basis change unitary $U_O$. The authors define a general output equation (Eq. 1) dependent on input state parameters and trainable weights.
• Architecture: The network consists of $K$ sequentially connected single-qubit neurons. The cumulative effect of these neurons is mathematically equivalent to a single effective quantum neuron.
• Data Encoding: Unlike previous approaches that decompose high-dimensional inputs into multiple 3D vectors (increasing circuit depth), this method employs a "data re-uploading" strategy. The input data is classically pre-processed to compute angles for the rotation gates. Specifically, the rotation parameters are modeled as additive polynomial functions of the input features (Eq. 4), allowing the network to handle arbitrary input dimensions directly.
• Reduced Model: A simplified version of the SQQNN is introduced where the input state and observable are fixed (measured in the computational basis), reducing the trainable parameters to a single angle $\beta$ per neuron, which sums to a global rotation angle.
• Training Strategies:
  • Regression: The model uses Gradient Descent to minimize the Mean Square Error (MSE). The authors provide analytical expressions for the partial derivatives of the output with respect to the network angles, facilitating efficient optimization.
  • Classification: For binary classification, the authors introduce a novel Polynomial-based Linear Least Squares (LLS) method. By transforming the classification problem into a regression task via an $\text{arctanh}$ function on the labels, the weights can be determined in a single step using the Moore–Penrose pseudo-inverse (Eq. 6). This approach seeks a global minimizer of the transformed quadratic least-squares objective, avoiding local minima common in iterative gradient-based methods.

Key Contributions

1. Resource-Efficient Architecture: The proposal of a Single-Qubit QNN that generalizes previous work (Ref. [31]) by handling multi-dimensional inputs without decomposing them, thereby reducing circuit depth and hardware demands.
2. Novel Training Algorithm: The development of a closed-form, matrix-based training method for classification inspired by polynomial regression. This method allows for rapid, deterministic parameter estimation in a single step, significantly accelerating training compared to iterative gradient descent.
3. Versatility: Demonstration of the SQQNN's capability to perform both regression (logic gates, continuous functions, real-world datasets) and binary classification (synthetic and real-world datasets) with high accuracy.

Results
The authors evaluated the SQQNN using classical simulations of quantum circuits on various datasets:

• Regression Tasks:
  • Logic Gates: The SQQNN achieved low training errors (MSE ranging from $6 \cdot 10^{-4}$ to $2 \cdot 10^{-3}$) for AND, OR, XOR, NAND, NOR, and XNOR gates with relatively few gradient descent epochs (38–79).
  • Sinc Function: The model successfully approximated the oscillatory sinc function, maintaining performance even with added white noise, converging to an MSE of approximately $10^{-3}$.
  • Real-World Datasets: On the Combined Cycle Power Plant (CCPP) dataset, the model achieved a test MSE of $0.0035 \pm 0.0002$, outperforming previous quantum results by an order of magnitude. On the Communities and Crime dataset, it achieved a test MSE of $0.1195 \pm 0.0252$, significantly lower than classical benchmarks cited in the literature.
• Classification Tasks:
  • Two Moons: The model reached 100% accuracy on this non-linearly separable synthetic dataset with a model order $K_{code} \ge 4$.
  • Wisconsin Breast Cancer (WBCD): The model achieved an accuracy of $0.956 \pm 0.028$, slightly outperforming a quantum bagging ensemble reported in the literature.
  • MNIST: Using a reduced SQQNN with $K_{code}=1$ and Discrete Cosine Transform (DCT) pre-processing, the model achieved near-perfect accuracy (e.g., $0.999 \pm 0.001$ for digits 0 vs. 1) on binary digit classification tasks. Training times were reported as approximately 8 seconds per iteration on a standard laptop.

Significance and Claims
The paper claims that the SQQNN offers a versatile, scalable, and practical alternative to multi-qubit models for near-term quantum devices. By relying on single-qubit gates, which are among the most reliable operations on current hardware with minimal decoherence, the model is well-suited for deployment on NISQ devices. The authors emphasize that the polynomial-based LLS training method provides a significant speedup for classification tasks by finding global minimizers of the transformed quadratic least-squares objective in a single step.

The results demonstrate that the SQQNN can learn both linear and non-linear relationships with strong performance in many tested scenarios. While the current results are based on classical simulations, the authors note that the shallow depth and single-qubit nature of the circuits make them favorable for real quantum hardware, though experimental validation on actual devices remains a necessary future step. The work underscores the potential of lightweight quantum models to bridge the gap between current quantum hardware capabilities and practical machine learning applications.