AI 中文总结
该研究针对量子搜索未充分利用希尔伯特空间结构的问题，提出基于希尔伯特空间递归分解的精确最优量子搜索方法，在非结构化搜索和网格空间搜索中均达到最优复杂度，为量子算法设计提供新途径。
AI 中文摘要
当前量子搜索方法未能充分挖掘底层希尔伯特空间中存在的结构，利用该结构对搜索进行分解可为量子搜索和算法设计提供新策略与新表述。我们提出一种直接作用于该结构的新分解技术，通过递归分解希尔伯特空间，并基于所得划分的反射构造搜索算子。当初始态和目标态可在该划分上分解时，每一层的动力学简化为二维平面内的单次旋转，旋转角度由标量递推关系给出。该递推关系避免了对每一层成功概率单独界定所产生的误差累积，从而得到精确的状态描述，可将递推作为整体处理。我们确定性地获得目标态，并推导得出与搜索设置无关的预言机（oracle）和非预言机代价。对于非结构化搜索，我们的方法同时达到最优的Θ(√N)预言机和非预言机门数；对于d维网格上的空间搜索，其在d≥3时恢复出O(√N)时间复杂度，在d=2时得到Aaronson与Ambainis提出的O(√N(log N)^(3/2))界。该递推的精确描述可扩展至我们的分解所覆盖的新细分结构，并为量子算法设计中递推的应用与分析提供新途径。
英文摘要
Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal $Θ(\sqrt{N})$ oracle and non-oracle gate counts. For spatial search on $d$-dimension grids, it recovers the $O(\sqrt{N})$ time for $d\geq3$ and the $O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)$ bound of Aaronson and Ambainis for $d=2$. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.
Comments28 pages, 3 figures
