  Hydrogen may be incorporated into nominally anhydrous minerals including
bridgmanite and post-perovskite as defects, making the Earth's deep mantle a
potentially significant water reservoir. The diffusion of hydrogen and its
contribution to the electrical conductivity in the lower mantle are rarely
explored and remain largely unconstrained. Here we calculate hydrogen
diffusivity in hydrous bridgmanite and post-perovskite, using molecular
dynamics simulations driven by machine learning potentials of ab initio
quality. Our findings reveal that hydrogen diffusivity significantly increases
with increasing temperature and decreasing pressure, and is considerably
sensitive to hydrogen incorporation mechanism. Among the four defect mechanisms
examined, (Mg + 2H)$_{\rm Si}$ and (Al + H)$_{\rm Si}$ show similar patterns
and yield the highest hydrogen diffusivity. Hydrogen diffusion is generally
faster in post-perovskite than in bridgmanite, and these two phases exhibit
distinct diffusion anisotropies. Overall, hydrogen diffusion is slow on
geological time scales and may result in heterogeneous water distribution in
the lower mantle. Additionally, the proton conductivity of bridgmanite for (Mg
+ 2H)$_{\rm Si}$ and (Al + H)$_{\rm Si}$ defects aligns with the same order of
magnitude of lower mantle conductivity, suggesting that the water distribution
in the lower mantle may be inferred by examining the heterogeneity of
electrical conductivity.
