Optimal Quantum Algorithms for Ordered Search

Ordered search is the problem of locating a target element in a sorted list of size n using comparison queries. Classically, binary search requires 2 n 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 π n+o(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.