  We develop a surface hopping algorithm based on frozen Gaussian approximation
for semiclassical matrix Schr\"odinger equations, in the spirit of Tully's
fewest switches surface hopping method. The algorithm is asymptotically derived
from the Schr\"odinger equation with rigorous approximation error analysis. The
resulting algorithm can be viewed as a path integral stochastic representation
of the semiclassical matrix Schr\"odinger equations. Our results provide
mathematical understanding to and shed new light on the important class of
surface hopping methods in theoretical and computational chemistry.
