Optimal Quantum Algorithms for Ordered Search

Abstract:Ordered search is the problem of locating a target element in a sorted list of size $n$ using comparison queries. Classically, binary search requires $\lceil \log_2 n\rceil$ queries, which is optimal. Quantum algorithms offer a constant-factor speedup, but the precise constant has been a longstanding open question. We close this gap by exhibiting two new quantum algorithms for ordered search, each using the optimal $\frac{1}{\pi}\ln n+o(\log n)$ queries. The first, discovered by Claude Fable 5, is a simple zero-error algorithm derived from a continuum relaxation of the problem. The second, discovered by GPT-5.6-Sol (informed by Claude's zero-error algorithm), is an exact algorithm based on an analytic solution of the polynomial program of Farhi, Goldstone, Gutmann, and Sipser.

arXiv:2609.34293 [quant-ph]

(or arXiv:2609.34293v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.34293 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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