Quantum algorithm for learning secret strings and its experimental demonstration

Abstract

In this paper, we consider the secret-string-learning problem in the teacher–student setting: the teacher has a secret string s∈{0,1}n, and the student wants to learn the secret s by question–answer interactions with the teacher, where at each time, the student can ask the teacher with a pair (x,q)∈{0,1}n×{0,1,…,n−1} and the teacher returns a bit given by the oracle fs(x,q) that indicates whether the length of the longest common prefix of s and x is greater than q or not. Our contributions are as follows.(i) We prove that any classical deterministic algorithm needs at least n queries to the oracle fs to learn the n-bit secret string s in both the worst case and the average case, and also present an optimal classical deterministic algorithm learning any s using n queries.(ii) We propose a quantum algorithm learning the n-bit secret string s with certainty using n/2 queries to the oracle fs, thus proving a double speedup over classical counterparts.(iii) Experimental demonstrations of our quantum algorithm on the IBM cloud quantum computer are presented, with the average success probabilities of 85.3% and 82.5% for all cases with n=2 and n=3, respectively.