  The state of a quantum system, adiabatically driven in a cycle, may acquire a
measurable phase depending only on the closed trajectory in parameter space.
Such geometric phases are ubiquitous, and also underline the physics of robust
topological phenomena such as the quantum Hall effect. Equivalently, a
geometric phase may be induced through a cyclic sequence of quantum
measurements. We show that the application of a sequence of weak measurements
renders the closed trajectories, hence the geometric phase, stochastic. We
study the concomitant probability distribution and show that, when varying the
measurement strength, the mapping between the measurement sequence and the
geometric phase undergoes a topological transition. Our finding may impact
measurement-induced control and manipulation of quantum states---a promising
approach to quantum information processing. It also has repercussions on
understanding the foundations of quantum measurement.

Quantum Physics;Mesoscale and Nanoscale Physics