Qubit-Efficient Quantum Search for Hyperdimensional Decomposition via Logarithmic Encoding

Hyperdimensional Computing (HDC) is a robust computational framework inspired by human cognition characterized by simple and efficient operations within high-dimensional vector spaces. Quantum-enhanced Hyperdimensional Computing (QeHDC) extends classical HDC by leveraging quantum mechanical properties to enhance computational efficiency. In this paper, we propose a novel Quantum HDC framework featuring a one-pass training method, leveraging sinusoidal and quantum encoding to project classical data into quantum amplitude states efficiently. Our framework introduces an innovative reference-state-based quantum binding operation realized via quantum circuits. Furthermore, we propose a density-matrix-based superclass generation strategy employing eigenvalue decomposition to extract critical quantum state features effectively, enabling a more accurate and robust class representation. Experimental evaluations conducted on standard benchmark datasets demonstrate our approach's superior performance, robustness to noise, and computational feasibility compared to traditional classical and existing quantum-enhanced approaches. The results highlight the practical benefits and potential of Quantum HDC for quantum-enhanced classification tasks and pave the way for future advancements in quantum-inspired computational paradigms.

Large language and sentence-embedding models provide rich semantic representations, but their high dimensionality poses a challenge for near-term quantum machine learning (QML), where quantum circuits can process only a limited number of input features. We investigate a hybrid quantum-classical pipeline that transforms high-dimensional sentence embeddings into compact representations for variational quantum classification. The workflow combines a pretrained sentence-embedding model, dimensionality reduction, angle encoding, a variational quantum circuit (VQC), and a classical decision layer. We systematically compare principal component analysis (PCA), neighborhood components analysis (NCA), and linear discriminant analysis (LDA), covering both unsupervised and supervised dimensionality reduction. Using the TREC question-classification dataset, we study the relationship between representation dimensionality, information retention, qubit count, and classification performance. Preliminary PCA experiments reveal a strong information bottleneck: reducing 768-dimensional embeddings to 3, 4, 5, and 8 dimensions retains about 8.2%, 10.2%, 11.9%, and 16.4% of the variance, with corresponding classification accuracies of 50.3%, 51.2%, 57.9%, and 63.4%. In contrast, supervised reduction is substantially more efficient. LDA reaches 85.3% accuracy and NCA reaches 83.1% using only 5 dimensions, under a leakage-free cross-validation protocol, compared with 85.1% for a full 384-dimensional classical baseline. These results indicate that supervised dimensionality reduction can preserve task-relevant information far more effectively than variance-based compression, making compact representations a promising route toward practical hybrid quantum-classical NLP models.

Hyperdimensional computing (HDC) is an approach from the cognitive science literature for solving information processing tasks using data represented as high-dimensional random vectors. The technique has a rigorous mathematical backing, and is easy to implement in energy-efficient and highly parallel hardware like FPGAs and "processing-in-memory" architectures. The effectiveness of HDC in machine learning largely depends on how raw data is mapped to high-dimensional space. In this work, we propose NysHD, a new method for constructing this mapping that is based on the Nyström method from the literature on kernel approximation. Our approach provides a simple recipe to turn any user-defined positive-semidefinite similarity function into an equivalent mapping in HDC. There is a vast literature on the design of such functions for learning problems. Our approach provides a mechanism to import them into the HDC setting, expanding the types of problems that can be tackled using HDC. Empirical evaluation against existing HDC encoding methods shows that NysHD can achieve, on average, 11% and 17% better classification accuracy on graph and string datasets respectively.