  Measurement plays a quintessential role in the control of quantum systems.
Beyond initialization and readout which pertain to projective measurements,
weak measurements in particular, through their back-action on the system, may
enable various levels of coherent control. The latter ranges from observing
quantum trajectories to state dragging and steering. Furthermore, just like the
adiabatic evolution of quantum states that is known to induce the Berry phase,
sequential weak measurements may lead to path-dependent geometric phases. Here
we measure the geometric phases induced by sequences of weak measurements and
demonstrate a topological transition in the geometric phase controlled by
measurement strength. This connection between weak measurement induced quantum
dynamics and topological transitions reveals subtle topological features in
measurement-based manipulation of quantum systems. Our protocol could be
implemented for classes of operations (e.g. braiding) which are topological in
nature. Furthermore, our results open new horizons for measurement-enabled
quantum control of many-body topological states.
