  In NMR (Nuclear Magnetic Resonance) quantum computation, the selective
control of multiple homonuclear spins is usually slow because their resonance
frequencies are very close to each other. To quickly implement controls against
decoherence effects, this paper presents an efficient numerical algorithm
fordesigning minimum-time local transformations in two homonuclear spins. We
obtain an accurate minimum-time estimation via geometric analysis on the
two-timescale decomposition of the dynamics. Such estimation narrows down the
range of search for the minimum-time control with a gradient-type optimization
algorithm. Numerical simulations show that this method can remarkably reduce
the search efforts, especially when the frequency difference is very small and
the control field is high. Its effectiveness is further demonstrated by NMR
experiments with two homunuclear carbon spins in a trichloroethylene (C2H1Cl3)
sample system.
