Quantum Search without Global Diffusion

Quantum Physics [Submitted on 16 Apr 2026] Quantum Search without Global Diffusion John Burke, Ciaran McGoldrick 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. Comments: 24 pages, 5 figures. Data and code available on request Subjects: Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS) MSC classes: 81P68 (Primary) 68Q12, 68Q25 (Secondary) ACM classes: F.1.2; F.2.2 Cite as: arXiv:2604.15435 [quant-ph]   (or arXiv:2604.15435v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.15435 Focus to learn more Submission history From: John Burke [view email] [v1] Thu, 16 Apr 2026 18:00:27 UTC (121 KB) Access Paper: view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.DS References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)