  We study spatial search with continuous-time quantum walks on real-world
complex networks. We use smaller replicas of the Internet network obtained with
a recent geometric renormalization method introduced by Garc\'ia-P\'erez et
al., Nat. Phys. 14, 583 (2018). This allows us to infer for the first time the
behavior of a quantum spatial search algorithm on a real-world complex network.
By simulating numerically the dynamics and optimizing the coupling parameter,
we study the optimality of the algorithm and its scaling with the size of the
network, showing that on average it is considerably better than the classical
scaling $\mathcal{O}(N)$, but it does not reach the ideal quadratic speedup
$\mathcal{O}(\sqrt{N})$ that can be achieved, e.g. in complete graphs. However,
the performance of the search algorithm strongly depends on the degree of the
nodes and, in fact, the scaling is found to be very close to optimal when we
consider the nodes below the $99$th percentile ordered according to the degree.
