  Understanding the escape of Lyman continuum (LyC) and Lyman $\alpha$
(Ly$\alpha$) photons from giant molecular clouds (GMCs) is crucial if we are to
study the reionization of the Universe and to interpret spectra of observed
galaxies at high redshift. To this end, we perform high-resolution,
radiation-magneto-hydrodynamic simulations of GMCs with self-consistent star
formation and stellar feedback. We find that a significant fraction (15-70%) of
ionizing radiation escapes from the simulated GMCs with different masses
($10^5$ and $10^6\,M_\odot$), as the clouds are dispersed within about
$2$-$5\,{\rm Myr}$ from the onset of star formation. The fraction of LyC
photons leaked is larger when the GMCs are less massive, metal-poor, less
turbulent, and less dense. The most efficient leakage of LyC radiation occurs
when the total star formation efficiency of a GMC is about 20%. The escape of
Ly$\alpha$ shows a trend similar to that of LyC photons, except that the
fraction of Ly$\alpha$ photons escaping from the GMCs is larger ($f_{\rm
esc}^{\rm Ly\alpha}\approx f_{900}^{0.27}$) and that a GMC with strong
turbulence shows larger $f_{\rm esc}^{\rm Ly\alpha}$. The simulated GMCs show a
characteristic velocity separation of $\Delta v\approx 120 \,{\rm km\,s^{-1}}$
in the time-averaged emergent Ly$\alpha$ spectra, suggesting that Ly$\alpha$
could be useful to infer the kinematics of the interstellar and circumgalactic
medium. We show that Ly$\alpha$ luminosities are a useful indicator of the LyC
escape, provided the number of LyC photons can be deduced through stellar
population modeling. Finally, we find that the correlations between the escape
fractions of Ly$\alpha$, ultraviolet photons at 1500A, and the Balmer $\alpha$
line are weak.
