  We present and analyze a protocol for driven-dissipatively preparing and
stabilizing a manifold of quantum manybody entangled states with
symmetry-protected topological order.
  Specifically, we consider 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 $S^\mathrm{total}=2$ subspace for each adjacent pair, and
thus, the stabilization of the manybody system into the Affleck, Kennedy, Lieb,
and Tasaki (AKLT) state up to the edge mode configuration. 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.
