  The problem of finding a marked node in a graph can be solved by the spatial
search algorithm based on continuous-time quantum walks (CTQW). However, this
algorithm is known to run in optimal time only for a handful of graphs. In this
work, we prove that for Erd\"os-Renyi random graphs, i.e.\ graphs of $n$
vertices where each edge exists with probability $p$, search by CTQW is
\textit{almost surely} optimal as long as $p\geq \log^{3/2}(n)/n$.
Consequently, we show that quantum spatial search is in fact optimal for
\emph{almost all} graphs, meaning that the fraction of graphs of $n$ vertices
for which this optimality holds tends to one in the asymptotic limit. We obtain
this result by proving that search is optimal on graphs where the ratio between
the second largest and the largest eigenvalue is bounded by a constant smaller
than 1. Finally, we show that we can extend our results on search to establish
high fidelity quantum communication between two arbitrary nodes of a random
network of interacting qubits, namely to perform quantum state transfer, as
well as entanglement generation. Our work shows that quantum information tasks
typically designed for structured systems retain performance in very disordered
structures.
