用对数编码大幅减少量子搜索的量子比特需求，实现高效高维分解。
Qubit-Efficient Quantum Search for Hyperdimensional Decomposition via Logarithmic Encoding
- 采用对数级编码替代传统高维编码，降低量子比特占用。
- 在保持平方根搜索复杂度的前提下，最多减少2000倍量子比特使用。
- 适合需要大规模高维计算且资源受限的量子信息处理场景。
高维计算（HDC）通过高维超向量（维度D）表示符号。在超向量分解中，目标是从一个绑定后的目标超向量中恢复出F个来自大小为N的码本的原始超向量，需搜索$N^F$个候选组合，计算成本极高。现有量子方法虽提供二次加速，但通常依赖$O(D)$量子比特的超向量表示，效率低下。本文提出一种量子比特高效的框架，将表示成本降至$O("log D$)，引入对数编码的超向量与绑定机制，并设计可逆的超向量查找算子，实现稠密超向量的电路级操作。结合改进的Dürr-Høyer搜索算法，该方法维持$O(\sqrt{N^F})$的搜索复杂度，同时显著降低量子比特消耗。实验验证了相似性计算正确性、可执行范围内的准确分解能力，以及相比基于显式$D$-量子比特编码的基线方案，最多提升2000倍的量子比特缩减效果。
原文摘要 · Abstract (English)
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.
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