2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/90686We consider a quantum gate, driven by a general time-dependent Hamiltonian, that complements the state of a qubit and then adds to it an arbitrary phase shift. It is shown that the minimum operation time of the gate is tau = (h/4E)(1+2 theta/pi), where h is Planck's constant, E is the average over time of the quantum-mechanical average energy, and theta is the phase shift modulo pi.5 page Latex w/ macro file latex-3.tex. Full paper for QS2002 (Quantum Structures) proceedings. This revision fixes two small typosQuantum Physicstext