  Building a network that is resilient to a component failure is vital. Our
access to electricity and telecommunications or the internet of things all
hinge on an uninterrupted service provided by a robust network. Calculating the
network reliability $R$ is $\sharp$P-complete and intractable to calculate
exactly for medium and large networks. Here, we present an explicit,
circuit-level implementation of a quantum algorithm that computes $R$. Our
algorithm requires $O(EV/\epsilon)$ gate operations and $O(E)$ qubits, where
$V$ and $E$ are the number of nodes and edges in the graph and $\epsilon$ is
the uncertainty in the reliability estimation. This constitutes a significant
polynomial speedup over the best classical approaches currently known. We
further provide quantum gate counts, relevant for both pre-fault-tolerant and
fault-tolerant regimes, sufficient to compute $R$.
