the experimental platform consisting of superconducting transmon circuits and linear microwave resonators. We perform theoretical modeling of this platform via pulse-level simulations based on physical features of real devices. In our protocol, transmon qutrits are mapped onto spin-1 systems. The qutrits‘ sharing of nearest-neighbor dispersive coupling to a dissipative microwave resonator enables elimination of state population in the Stotal = 2 subspace for each adjacent pair, and thus, the stabilization of the manybody system into the Affleck, Kennedy, Lieb and Tasaki (AKLT) state. We also analyze the performance of our protocol as the system size scales up to four qutrits, in terms of its fidelity as well as the stabilization time. Our work shows the capacity of driven-dissipative superconducting cQED systems to host robust and self-corrected quantum manybody states that are topologically non-trivial.

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.