  We study the effects of doping the Kitaev model on the honeycomb lattice
where the spins interact via the bond-directional interaction $J_K$, which is
known to have a quantum spin liquid as its exact ground state. The effect of
hole doping is studied within the $t$-$J_K$ model on a three-leg cylinder using
density-matrix renormalization group. Upon light doping, we find that the
ground state of the system has quasi-long-range charge-density-wave
correlations but short-range single-particle correlations. The dominant pairing
channel is the even-parity superconducting pair-pair correlations with
$d$-wave-like symmetry, which oscillate in sign as a function of separation
with a period equal to that of the spin-density wave and two times the
charge-density wave. Although these correlations fall rapidly (possibly
exponentially) at long distances, this is never-the-less the first example
where a pair-density wave is the strongest SC order on a bipartite lattice. Our
results may be relevant to ${\rm Na_2IrO_3}$ and $\alpha$-${\rm RuCl_3}$ upon
doping.
