Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the $O(\sqrt{N})$ oracle complexity of Grover search when each partition contains at least $\log_2(\log_2 N)$ qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.

翻译：量子搜索是量子计算中最重要的算法之一。其核心是量子振幅放大技术，通过结合两种全局反射操作（标记目标的Oracle算子和关于初始态进行反射的扩散算子）来实现相对于经典搜索的二次加速。我们证明，当Oracle是唯一全局算子，而所有其他操作仅作用于搜索寄存器非重叠分区上的局部操作时，这种加速仍可保持。我们提出一种递归构建方案，当初始态和目标态均能在这些选定分区上分解为张量积形式时，该方案可获得算法动力学的精确闭式解。这一成果得益于连续反射间主角度的有趣简并现象——这些角度坍缩为仅由单个递归定义角度支配的两个不同值。当应用于自然满足张量分解条件的非结构化搜索时，若每个分区包含至少$\log_2(\log_2 N)$个量子比特，该方法可保留Grover搜索的$O(\sqrt{N})$次Oracle调用复杂度。在18量子比特的搜索问题中，采用两阶段划分相较于Grover算法可将非Oracle电路深度降低51%-96%，仅需额外增加9%的Oracle调用次数。对于更大规模问题，这种Oracle开销会迅速减少，而当Oracle电路深度显著大于扩散算子时，深度缩减效益依然显著。更广泛地说，这些结果表明全局扩散算子并非实现量子搜索二次加速的必要条件，为这种基础性算法提供了全新视角。此外，我们分析核心的标量约化方法，为量子算法设计与评估启发了新的研究方向与创新实践。