  We analyze the physics of optimal protocols to prepare a target state with
high fidelity in a symmetrically coupled two-qubit system. By varying the
protocol duration, we find a discontinuous phase transition, which is
characterized by a spontaneous breaking of a $\mathbb{Z}_2$ symmetry in the
functional form of the optimal protocol, and occurs below the quantum speed
limit. We study in detail this phase and demonstrate that even though
high-fidelity protocols come degenerate with respect to their fidelity, they
lead to final states of different entanglement entropy shared between the
qubits. Consequently, while globally both optimal protocols are equally far
away from the target state, one is locally closer than the other. An
approximate variational mean-field theory which captures the physics of the
different phases is developed.

Statistical Mechanics;Other Condensed Matter;Quantum Gases