  In this paper, we explore quantum speedups for the problem, inspired by
matroid theory, of identifying 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 $s, s'\in\{0, 1\}^n$ satisfying the above constraints, for any
$x\in\{0, 1\}^n$ the max inner product oracle $O_{max}(x)$ returns the max
value between $s\cdot x$ and $s'\cdot x$, and the sub-set oracle $O_{sub}(x)$
indicates whether the index set of the 1s in $x$ is a subset of that in $s$ or
$s'$. We present a quantum algorithm consuming $O(1)$ queries to the max inner
product oracle for identifying the pair $\{s, s'\}$, and prove that any
classical algorithm requires $\Omega(n/\log_{2}n)$ queries. Also, we present a
quantum algorithm consuming $\frac{n}{2}+O(\sqrt{n})$ queries to the subset
oracle, and prove that any classical algorithm requires at least $n+\Omega(1)$
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.
