  Quantum theory allows the traversing of multiple channels in a superposition
of different orders. When the order in which the channels are traversed is
controlled by an auxiliary quantum system, various unknown parameters of the
channels can be estimated by measuring only the control system, even when the
state of the probe alone would be insensitive. Moreover, increasing the
dimension of the control system increases the number of simultaneously
estimable parameters, which has important metrological ramifications. We
demonstrate this capability for simultaneously estimating both unitary and
noise parameters, including multiple parameters from the same unitary such as
rotation angles and axes and from noise channels such as depolarization,
dephasing, and amplitude damping in arbitrary dimensions. We identify regimes
of unlimited advantages, taking the form of $p^2$ smaller variances in
estimation when the noise probability is $1-p$, for both single and
multiparameter estimation when using our schemes relative to any comparable
scheme whose causal order is definite.
