  We address continuous-time quantum walks on graphs, and discuss whether and
how quantum-limited measurements on the walker may extract information on the
tunnelling amplitude between the nodes of the graphs. For a few remarkable
families of graphs, we evaluate the ultimate quantum bound to precision, i.e.
we compute the quantum Fisher information (QFI), and assess the performances of
incomplete measurements, i.e. measurements performed on a subset of the graph's
nodes. We also optimize the QFI over the initial preparation of the walker and
find the optimal measurement achieving the ultimate precision in each case. As
the topology of the graph is changed, a non-trivial interplay between the
connectivity and the achievable precision is uncovered.

Quantum Physics