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Understanding the escape of Lyman continuum (LyC) and Lyman $\alpha$(Ly$\alpha$) photons from giant molecular clouds (GMCs) is crucial if we are tostudy the reionization of the Universe and to interpret spectra of observedgalaxies at high redshift. To this end, we perform high-resolution,radiation-magneto-hydrodynamic simulations of GMCs with self-consistent starformation and stellar feedback. We find that a significant fraction (15-70%) ofionizing 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 LyCphotons leaked is larger when the GMCs are less massive, metal-poor, lessturbulent, and less dense. The most efficient leakage of LyC radiation occurswhen the total star formation efficiency of a GMC is about 20%. The escape ofLy$\alpha$ shows a trend similar to that of LyC photons, except that thefraction of Ly$\alpha$ photons escaping from the GMCs is larger ($f_{\rmesc}^{\rm Ly\alpha}\approx f_{900}^{0.27}$) and that a GMC with strongturbulence shows larger $f_{\rm esc}^{\rm Ly\alpha}$. The simulated GMCs show acharacteristic 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 circumgalacticmedium. We show that Ly$\alpha$ luminosities are a useful indicator of the LyCescape, provided the number of LyC photons can be deduced through stellarpopulation modeling. Finally, we find that the correlations between the escapefractions of Ly$\alpha$, ultraviolet photons at 1500A, and the Balmer $\alpha$line are weak.

A systematic study of the escape of LyC and Ly$\alpha$ photons from star-forming, magnetized turbulent clouds