  The matroid intersection problem is a fundamental problem that has been
extensively studied for half a century. In the classic version of this problem,
we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and
$\mathcal{M}_2 = (V, \mathcal{I}_2)$ on a comment ground set $V$ of $n$
elements, and then we have to find the largest common independent set $S \in
\mathcal{I}_1 \cap \mathcal{I}_2$ by making independence oracle queries of the
form "Is $S \in \mathcal{I}_1$?" or "Is $S \in \mathcal{I}_2$?" for $S
\subseteq V$. The goal is to minimize the number of queries.
  Beating the existing $\tilde O(n^2)$ bound, known as the quadratic barrier,
is an open problem that captures the limits of techniques from two lines of
work. The first one is the classic Cunningham's algorithm [SICOMP 1986], whose
$\tilde O(n^2)$-query implementations were shown by CLS+ [FOCS 2019] and Nguyen
[2019]. The other one is the general cutting plane method of Lee, Sidford, and
Wong [FOCS 2015]. The only progress towards breaking the quadratic barrier
requires either approximation algorithms or a more powerful rank oracle query
[CLS+ FOCS 2019]. No exact algorithm with $o(n^2)$ independence queries was
known.
  In this work, we break the quadratic barrier with a randomized algorithm
guaranteeing $\tilde O(n^{9/5})$ independence queries with high probability,
and a deterministic algorithm guaranteeing $\tilde O(n^{11/6})$ independence
queries. Our key insight is simple and fast algorithms to solve a graph
reachability problem that arose in the standard augmenting path framework
[Edmonds 1968]. Combining this with previous exact and approximation algorithms
leads to our results.
