  Utilizing optimal control to simulate a model Hamiltonian is an emerging
strategy that leverages the intrinsic physics of a device with digital quantum
simulation methods. Here we evaluate optimal control for probing the
non-equilibrium properties of symmetry-protected topological (SPT) states
simulated with superconducting hardware. Assuming a tunable transmon
architecture, we cast evolution of these SPT states as a series of one- and
two-site pulse optimization problems that are solved in the presence of leakage
constraints. From the generated pulses, we numerical simulate time-dependent
melting of the perturbed SPT string order across a six-site model with an
average state infidelity of $10^{-3}$. The feasibility of these pulses as well
as their efficient application indicate that high-fidelity simulations of
string-order melting are within reach of current quantum computing systems.
