The Bayesian Learner is Optimal for Noisy Binary Search (and Pretty Good for Quantum as Well)

We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants: • Each comparison is erroneous with independent probability 1− p. • At each stage k comparisons can be performed in parallel and a noisy answer is returned. We present a (classical) algorithm which solves both variants optimally (with respect to p and k), up to an additive term of O(loglog n), and prove matching information-theoretic lower bounds. We use the algorithm to improve the results of Farhi et al. [12], presenting an exact quantum search algorithm in an ordered list of expected complexity less than (log2 n)/3.