Single-Qudit Quantum Neural Networks for Multiclass Classification

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.