Approaching the quantum speed limit in quantum gates with geometric control

We present a geometric optimization method for implementing quantum gates by optimally controlling the Hamiltonian parameters, with the goal of approaching the Mandelstam-Tamm quantum speed limit (MT-QSL). Achieving this bound requires satisfying precise geometric conditions that govern the evolution of quantum states. We extend this geometric framework to quantum unitary operators in arbitrary dimensions and analyze the conditions necessary for saturation of the bound. Additionally, we show that the quantum brachistochrone, when generalized to operators, does not, in general, saturate the MT-QSL bound. Finally, we propose a systematic optimal control strategy based on geometric principles to approach the quantum speed limit for unitary driving. We illustrate this optimization method on a set of four well-known two-qubit quantum gates. Our procedure significantly reduces the deviation from the optimal quantum speed limit while preserving high quantum fidelity.