Quantum algorithms for learning hidden strings with applications to matroid problems

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of learning a pair of n-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string s, s ' is an element of {0, 1}n satisfying the above constraints, for any x is an element of {0, 1}n the max inner product oracle O,,,"x(x) returns the max value between s center dot x and s ' center dot x, and the sub-set oracle Os,iota 6(x) indicates whether the index set of the 1s in x is a subset of that in s or s '. We present an exact quantum algorithm consuming O(1) queries to the max inner product oracle for identifying the pair {s, s '}, and prove that any randomized algorithm requires omega(n/ log2 n) queries. Also, we root present a quantum algorithm consuming n 2 +O( n) queries to the subset oracle, and prove that any randomized algorithm requires at least n -3 + omega(log2 n) queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called k-bases if it has k bases.