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

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.

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