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arXiv · 2609.34293

Optimal Quantum Algorithms for Ordered Search

Joseph Carolan·Andrew M. Childs
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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 $\frac{1}π\ln n+o(\log n)$ 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.

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Joseph Carolan, Andrew M. Childs. 2026-09-28. Optimal Quantum Algorithms for Ordered Search. https://arxiv.org/abs/2609.34293

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