  Deconfined quantum criticality (DQC) arises from fractionalization of
quasi-particles and leads to fascinating behaviors beyond the
Landau-Ginzburg-Wilson (LGW) description of phase transitions. The unusual
aspects of DQC in equilibrium also suggest exotic phenomena out of equilibrium,
which are still largely unexplored. Here we study manifestations of DQC when
driving (quantum annealing) a two-dimensional quantum magnet through a critical
point separating antiferromagnetic and spontaneously dimerized ground states.
Numerical simulations show that the celebrated Kibble-Zurek scaling (KZS)
mechanism is inadequate for describing the annealing process. To explain our
results, we introduce the concept of dual asymmetric KZS, where a deconfinement
time enters in addition to the conventional relaxation time and the scaling
also depends on the direction in which the system is driven through the
critical point according to a duality principle connecting the topological
defects in the two phases. These defects -- spinons in the dimerized phase and
space-time hedgehogs in the antiferromagnetic phase -- require a much longer
time scale for equilibration than the amplitude of the order parameter. Beyond
the new insights into DQC, our scaling approach provides a new window into
out-of-equilibrium criticality with multiple length and time scales.
