  A generic qubit unitary operator affected by quantum noise is duplicated and
inserted in a coherently superposed channel, superposing two paths offered to a
probe qubit across the noisy unitary, and driven by a control qubit. A
characterization is performed of the transformation realized by the superposed
channel on the joint state of the probe-control qubit pair. The superposed
channel is then specifically analyzed for the fundamental metrological task of
phase estimation on the noisy unitary, with the performance assessed by the
Fisher information, classical or quantum. A comparison is made with
conventional estimation techniques and also with a quantum switched channel
with indefinite causal order recently investigated for a similar task of phase
estimation. In the analysis here, a first important observation is that the
control qubit of the superposed channel, although it never directly interacts
with the unitary being estimated, can nevertheless be measured alone for
effective estimation, while discarding the probe qubit that interacts with the
unitary. This property is also present with the switched channel but is
inaccessible with conventional techniques. The optimal measurement of the
control qubit here is characterized in general conditions. A second important
observation is that the noise plays an essential role in coupling the control
qubit to the unitary, and that the control qubit remains operative for phase
estimation at very strong noise, even with a fully depolarizing noise, whereas
conventional estimation and the switched channel become inoperative in these
conditions. The results extend the analysis of the capabilities of coherently
controlled channels which represent novel devices exploitable for quantum
signal and information processing.
