Quantum Algorithms for Highly Structured Search Problems

We consider the problem of identifying a base k string given an oracle which returns information about the number of correct components in a query, specifically, the Hamming distance between the query and the solution, modulo r = max{2, 6 − k}. Classically this problem requires Ω(n logr k) queries. For k ∈ {2, 3, 4} we construct quantum algorithms requiring only a single quantum query. For k > 4, we show that O( √ k) quantum queries suffice. In both case, the quantum algorithms are optimal. 2001 Physics and Astronomy Classification Scheme: 03.67.Lx. 2000 American Mathematical Society Subject Classification: 81P68, 68Q17.