  We demonstrate the efficiency of a recent exact-gradient optimal control
methodology by applying it to a challenging many-body problem, crossing the
superfluid to Mott-insulator phase transition in the Bose-Hubbard model. The
system size necessitates a matrix product state representation and this
seamlessly integrates with the requirements of the algorithm. We observe
fidelities in the range 0.99-0.9999 with associated minimal process duration
estimates displaying an exponential fidelity-duration trade-off across several
orders of magnitude. The corresponding optimal solutions are characterized in
terms of a predominantly linear sweep across the critical point followed by
bang-bang-like structure. This is quite different from the smooth and monotonic
solutions identified by earlier gradient-free optimizations which are hampered
in locating the higher complexity protocols in the regime of high-fidelities at
low process durations. Overall, the comparison suggests significant
methodological improvements also for many-body systems in the ideal open-loop
setting. Acknowledging that idealized open-loop control may deteriorate in
actual experiments, we discuss the merits of using such an approach in
combination with closed-loop control -- in particular, high-fidelity physical
insights extracted with the former can be used to formulate practical,
low-dimensional search spaces for the latter.
