  The advantage of quantum metrology has been experimentally demonstrated for
phase estimations where the dynamics are commuting. General noncommuting
dynamics, however, can have distinct features. For example, the direct
sequential scheme, which can achieve the Heisenberg scaling for the phase
estimation under commuting dynamics, can have even worse performances than the
classical scheme under noncommuting dynamics. Here we realize a scalable
optimally controlled sequential scheme, which can achieve the Heisenberg
precision under general noncommuting dynamics. We also present an intuitive
geometrical framework for the controlled scheme and identify sweet spots in
time at which the optimal controls used in the scheme can be pre-fixed without
adaptation, which simplifies the experimental protocols significantly. We
successfully implement the scheme up to eight controls in an optical platform,
demonstrate a precision near the Heisenberg limit. Our work opens the avenue
for harvesting the power of quantum control in quantum metrology, and provides
a control-enhanced recipe to achieve the Heisenberg precision under general
noncommuting dynamics.
