Quantum algorithms for learning hidden strings with applications to matroid problems

Abstract

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′∈{0,1}n satisfying the above constraints, for any x∈{0,1}n the max inner product oracle Omax(x) returns the max value between s⋅x and s′⋅x, and the sub-set oracle Osub(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 Ω(n/log2⁡n) queries. Also, we present a quantum algorithm consuming n2+O(n) queries to the subset oracle, and prove that any randomized algorithm requires at least n−3+Ω(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.