  The fundamental problem of sampling from the limiting distribution of quantum
walks on networks, known as \emph{mixing}, finds widespread applications in
several areas of quantum information and computation. Of particular interest in
most of these applications, is the minimum time beyond which the instantaneous
probability distribution of the quantum walk remains close to this limiting
distribution, known as the \emph{quantum mixing time}. However this quantity is
only known for a handful of specific networks. In this letter, we prove an
upper bound on the quantum mixing time for \emph{almost all networks}, i.e.\
the fraction of networks for which our bound holds, goes to one in the
asymptotic limit. To this end, using several results in random matrix theory,
we find the quantum mixing time of Erd\"os-Renyi random networks: networks of
$n$ nodes where each edge exists with probability $p$ independently. For
example for dense random networks, where $p$ is a constant, we show that the
quantum mixing time is $\mathcal{O}\left(n^{3/2 + o(1)}\right)$. Besides
opening avenues for the analytical study of quantum dynamics on random
networks, our work could find applications beyond quantum information
processing. Owing to the universality of Wigner random matrices, our results on
the spectral properties of random graphs hold for general classes of random
matrices that are ubiquitous in several areas of physics. In particular, our
results could lead to novel insights into the equilibration times of isolated
quantum systems defined by random Hamiltonians, a foundational problem in
quantum statistical mechanics.
