  We derive the explicit solution of the problem of time-optimal control by a
common magnetic fields for two independent spin-$\frac{1}{2}$ particles. Our
approach is based on the Pontryagin Maximum Principle and a novel symmetry
reduction technique. We experimentally implement the optimal control using
zero-field nuclear magnetic resonance. This reveals an average gate error of
$1\%$ and a $70 \%$ to $80$ $\%$ decrease in the experiment duration as
compared to existing methods. This is the first analytical solution and
experimental demonstration of time-optimal control in such a system and it
provides a route to achieve time optimal control in more general quantum
systems.
