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We study how useful random states are for quantum metrology, i.e., surpassthe classical limits imposed on precision in the canonical phase estimationscenario. First, we prove that random pure states drawn from the Hilbert spaceof distinguishable particles typically do not lead to super-classical scalingof precision even when allowing for local unitary optimization. Conversely, weshow that random states from the symmetric subspace typically achieve theoptimal Heisenberg scaling without the need for local unitary optimization.Surprisingly, the Heisenberg scaling is observed for states of arbitrarily lowpurity and preserved under finite particle losses. Moreover, we prove that forsuch states a standard photon-counting interferometric measurement suffices totypically achieve the Heisenberg scaling of precision for all possible valuesof the phase at the same time. Finally, we demonstrate that metrologicallyuseful states can be prepared with short random optical circuits generated fromthree types of beam-splitters and a non-linear (Kerr-like) transformation.

Random bosonic states for robust quantum metrology