  In this work, we address the challenge of uncovering patterns in variational
optimal protocols for taking the system to ground states of many-body
Hamiltonians, using variational quantum algorithms. We develop highly optimized
classical Monte Carlo (MC) algorithms to find the optimal protocols for
transformations between the ground states of the square-lattice XXZ model for
finite systems sizes. The MC method obtains optimal bang-bang protocols, as
predicted by Pontryagin's minimum principle. We identify the minimum time
needed for reaching an acceptable error for different system sizes as a
function of the initial and target states and uncover correlations between the
total time and the wave-function overlap. We determine a dynamical phase
diagram for the optimal protocols, with different phases characterized by a
topological number, namely the number of on-pulses. Bifurcation transitions as
a function of initial and final states, associated with new jumps in the
optimal protocols, demarcate these different phases. The number of pulses
correlates with the total evolution time. In addition to identifying the
topological characteristic above, i.e., the number of pulses, we introduce a
correlation function to characterize bang-bang protocols' quantitative
geometric similarities. We find that protocols within one phase are indeed
geometrically correlated. Identifying and extrapolating patterns in these
protocols may inform efficient large-scale simulations on quantum devices.
