Qubit-Efficient Quantum Search for Hyperdimensional Decomposition via Logarithmic Encoding

2026-07-11

AI Summary

This work addresses the high computational cost of recovering multiple constituent components from a bound hypervector in high-dimensional computing, which classically requires searching a space of size $N^F$. While existing quantum approaches represent hypervectors using $O(D)$ qubits—leading to poor resource efficiency—we propose a quantum-efficient framework based on logarithmic encoding that compresses hypervectors to $O(\log D)$ qubits. By integrating a reversible hypervector lookup operator with an improved Dürr–Høyer quantum search algorithm, our method maintains the optimal $O(\sqrt{N^F})$ search complexity while drastically reducing qubit overhead. Experimental results confirm the accuracy of component decomposition and similarity computation, achieving up to a 2000-fold reduction in qubit requirements at feasible scales, thereby overcoming the longstanding $O(D)$ representation bottleneck.

Abstract

Hyperdimensional Computing (HDC) represents symbols using high-dimensional hypervectors of dimension $D$. In hypervector decomposition, the objective is to recover $F$ constituent hypervectors, each drawn from a codebook of size $N$, from a bound target hypervector. This requires searching over $N^F$ candidate tuples, making the task computationally prohibitive at scale. Recent quantum approach provides a quadratic search advantage, but typically rely on qubit-inefficient $O(D)$-qubit hypervector representations. We propose a qubit-efficient quantum framework for HDC decomposition that reduces the representation cost to $O(\log D)$. The framework introduces logarithmic hypervector and binding encodings, together with a reversible hypervector lookup operator for circuit-level manipulation of dense hypervectors. Combined with a modified Dürr-Høyer search procedure, the method preserves $O(\sqrt{N^F})$ search complexity while substantially reducing qubit usage. Experimental results validate correct similarity computation, accurate decomposition in executable regimes, and significantly improved qubit scaling over baselines based on explicit $D$-qubit hypervector encodings, achieving up to $2{,}000\times$ fewer qubits.

Problem

• Hyperdimensional Computing
• Quantum Search
• Qubit Efficiency
• Hypervector Decomposition
• Logarithmic Encoding

Innovation

• qubit-efficient
• logarithmic encoding
• hyperdimensional computing
• quantum search
• reversible lookup