  We address quantum spatial search on graphs and its implementation by
continuous-time quantum walks in the presence of dynamical noise. In
particular, we focus on search on the complete graph and on the star graph of
order $N$, proving that also the latter is optimal in the computational limit
$N \gg 1$, being nearly optimal also for small $N$. The noise is modeled by
independent sources of random telegraph noise (RTN), dynamically perturbing the
links of the graph. We observe two different behaviours depending on the
switching rate of RTN: fast noise only slightly degrades performance, whereas
slow noise is more detrimental and, in general, lowers the success probability.
In particular, we still find a quadratic speed-up for the average running time
of the algorithm, while for the star graph with external target node we observe
a transition to classical scaling. We also address how the effects of noise
depend on the order of the graphs, and discuss the role of the graph topology.
Overall, our results suggest that realizations of quantum spatial search are
possible with current technology, and also indicate the star graph as the
perfect candidate for the implementation by noisy quantum walks, owing to its
simple topology and nearly optimal performance also for just few nodes.
