Quantum Search in an Ordered List via Adaptive Learning

Abstract

We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants: 1. Each comparison can be erroneous with some probability $1 - p$. 2. At each stage $k$ comparisons can be performed in parallel and a noisy answer is returned We present a (classic) algorithm which optimally solves both variants together, up to an additive term of O(\log \log(n)), and prove matching information theoretic lower bounds. We use the algorithm to improve the results of Farhi et al \cite{FGGS99} presenting a quantum (error free) search algorithm in an ordered list of expected complexity less than (\log_2n) / 3.