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Single-Qudit Quantum Neural Networks for Multiclass Classification

Leandro C. Souza, Renato Portugal

TL;DR

This work introduces a single-qudit quantum neural network (QNN) for multiclass classification, leveraging a d-dimensional qudit and a Cayley-transform-based unitary to directly map class labels to measurement outcomes. A hybrid training pipeline combines a nonlinear activation based on a degree-L multivariable Taylor expansion with regularized hinge-loss SVM optimization, enabling efficient learning in a compact quantum circuit. Empirical evaluation on EMNIST and MNIST subsets demonstrates competitive accuracy (e.g., up to 98.88% on EMNIST Digits and 98.20% on EMNIST MNIST) while highlighting trade-offs between input dimensionality, network depth, and compute time. The study highlights the potential of qudit-based QNNs for scalable multiclass tasks, while acknowledging current hardware limitations and outlining pathways toward hardware-compatible implementations and extensions to regression tasks.

Abstract

This paper proposes a single-qudit quantum neural network for multiclass classification, by using the enhanced representational capacity of high-dimensional qudit states. Our design employs an $d$-dimensional unitary operator, where $d$ corresponds to the number of classes, constructed using the Cayley transform of a skew-symmetric matrix, to efficiently encode and process class information. This architecture enables a direct mapping between class labels and quantum measurement outcomes, reducing circuit depth and computational overhead. To optimize network parameters, we introduce a hybrid training approach that combines an extended activation function -- derived from a truncated multivariable Taylor series expansion -- with support vector machine optimization for weight determination. We evaluate our model on the MNIST and EMNIST datasets, demonstrating competitive accuracy while maintaining a compact single-qudit quantum circuit. Our findings highlight the potential of qudit-based QNNs as scalable alternatives to classical deep learning models, particularly for multiclass classification. However, practical implementation remains constrained by current quantum hardware limitations. This research advances quantum machine learning by demonstrating the feasibility of higher-dimensional quantum systems for efficient learning tasks.

Single-Qudit Quantum Neural Networks for Multiclass Classification

TL;DR

This work introduces a single-qudit quantum neural network (QNN) for multiclass classification, leveraging a d-dimensional qudit and a Cayley-transform-based unitary to directly map class labels to measurement outcomes. A hybrid training pipeline combines a nonlinear activation based on a degree-L multivariable Taylor expansion with regularized hinge-loss SVM optimization, enabling efficient learning in a compact quantum circuit. Empirical evaluation on EMNIST and MNIST subsets demonstrates competitive accuracy (e.g., up to 98.88% on EMNIST Digits and 98.20% on EMNIST MNIST) while highlighting trade-offs between input dimensionality, network depth, and compute time. The study highlights the potential of qudit-based QNNs for scalable multiclass tasks, while acknowledging current hardware limitations and outlining pathways toward hardware-compatible implementations and extensions to regression tasks.

Abstract

This paper proposes a single-qudit quantum neural network for multiclass classification, by using the enhanced representational capacity of high-dimensional qudit states. Our design employs an -dimensional unitary operator, where corresponds to the number of classes, constructed using the Cayley transform of a skew-symmetric matrix, to efficiently encode and process class information. This architecture enables a direct mapping between class labels and quantum measurement outcomes, reducing circuit depth and computational overhead. To optimize network parameters, we introduce a hybrid training approach that combines an extended activation function -- derived from a truncated multivariable Taylor series expansion -- with support vector machine optimization for weight determination. We evaluate our model on the MNIST and EMNIST datasets, demonstrating competitive accuracy while maintaining a compact single-qudit quantum circuit. Our findings highlight the potential of qudit-based QNNs as scalable alternatives to classical deep learning models, particularly for multiclass classification. However, practical implementation remains constrained by current quantum hardware limitations. This research advances quantum machine learning by demonstrating the feasibility of higher-dimensional quantum systems for efficient learning tasks.

Paper Structure

This paper contains 15 sections, 1 theorem, 31 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Qudit-Based Neuron Model
  3. Single-Qudit Quantum Neural Network
  4. Training Method
  5. Activation Function Design
  6. Weight Determination
  7. Class Determination
  8. Classification of EMNIST Datasets
  9. Balanced Dataset
  10. Digits Dataset
  11. Letters Dataset
  12. EMNIST MNIST Dataset
  13. Comparison
  14. Classification Performance on MNIST Dataset
  15. Conclusion and Future Work

Key Result

Lemma 1

Using the notation $c_{\ell,d}$ and $s_{\ell,d}$ to explicitly indicate the dimension of the Hilbert space, we have

Figures (3)

  • Figure 1: Illustration of a qudit-based quantum neuron. The input can be an arbitrary $d$-dimensional quantum state ${\left\vert{\psi}\right\rangle}_d$, which undergoes a transformation via the parameterized unitary gate $U(\vec{\theta}\,)$. A measurement in the computational basis produces one of $d$ possible outcomes, corresponding to the neuron's prediction.
  • Figure 2: Depiction of a single-qudit quantum neural network with $L$ layers. The input is a $d$-dimensional quantum state ${\left\vert{\psi}\right\rangle}_d$, which is sequentially transformed by parameterized unitary gates $U_\ell(\vec{\theta}^{(\ell)})$ for $\ell = 1, \cdots, L$. A measurement in the computational basis at the end of the network produces one of $d$ possible outcomes, corresponding to the network's prediction.
  • Figure 3: Example of a qubit-based circuit implementing the 5-dimensional unitary $U(\theta_1, \theta_2, \theta_3, \theta_4)$ described in Sec. \ref{['sec-neuron']} using multi-controlled $R_y$ gates.

Theorems & Definitions (2)

  • Lemma 1
  • proof