  Motivated by the recent experimental discovery of strongly
surface-plane-dependent superconductivity at surfaces of KTaO$_3$ single
crystals, we calculate the electron-phonon coupling strength, $\lambda$, of
doped KTaO$_3$ along the reciprocal-space high-symmetry directions. Using the
Wannier-function approach implemented in the EPW package, we calculate
$\lambda$ across the experimentally covered doping range and compare its
mode-resolved distribution along the [001], [110] and [111] reciprocal-space
directions. We find that the electron-phonon coupling is strongest in the
optical modes around the $\Gamma$ point, with some distribution to higher $k$
values in the [001] direction. The electron-phonon coupling strength as a
function of doping has a dome-like shape in all three directions and its
integrated total is largest in the [001] direction and smallest in the [111]
direction, in contrast to the experimentally measured trends in critical
temperatures. This disagreement points to a non-BCS character of the
superconductivity. Instead, the strong localization of $\lambda$ in the soft
optical modes around $\Gamma$ suggests an importance of ferroelectric soft-mode
fluctuations, which is supported by our findings that the mode-resolved
$\lambda$ values are strongly enhanced in polar structures. The inclusion of
spin-orbit coupling has negligible influence on our calculated mode-resolved
$\lambda$ values.
