Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: application to the XXZ model on the square lattice

Uncovering patterns in variational optimal protocols for taking a quantum system to ground states of many-body Hamiltonians is a significant challenge. Here, we develop highly efficient algorithms to find the optimal protocols for transformations between the ground states of the square-lattice XXZ model, which obtain 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 for various initial and target states and reveal 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 furthermore correlates with the total evolution time. In addition to the topological characteristic above, 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 noisy intermediate-scale quantum devices.