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We address continuous-time quantum walks on graphs, and discuss whether andhow quantum-limited measurements on the walker may extract information on thetunnelling amplitude between the nodes of the graphs. For a few remarkablefamilies of graphs, we evaluate the ultimate quantum bound to precision, i.e.we compute the quantum Fisher information (QFI), and assess the performances ofincomplete measurements, i.e. measurements performed on a subset of the graph'snodes. We also optimize the QFI over the initial preparation of the walker andfind the optimal measurement achieving the ultimate precision in each case. Asthe topology of the graph is changed, a non-trivial interplay between theconnectivity and the achievable precision is uncovered.

The walker speaks its graph: global and nearly-local probing of the tunnelling amplitude in continuous-time quantum walks