  The Kitaev-Heisenberg model on the honeycomb lattice has been studied for the
purpose of finding exotic states such as quantum spin liquid and topological
orders. On the kagome lattice, in spite of a spin-liquid ground state in the
Heisenberg model, the stability of the spin-liquid state has hardly been
studied in the presence of the Kitaev interaction. Therefore, we investigate
the ground state of the classical and quantum spin systems of the kagome
Kitaev-Heisenberg model. In the classical system, we obtain an exact phase
diagram that has an eight-fold degenerated canted ferromagnetic phase and a
subextensive degenerated Kitaev antiferromagnetic phase. In the quantum system,
using the Lanczos-type exact diagnalization and cluster mean-field methods, we
obtain two quantum spin-liquid phases, an eight-fold degenerated canted
ferromagnetic phase similar to the classical spin system, and an eight-fold
degenerated $\bf q=0$ $120^\circ$ ordered phase induced by quantum fluctuation.
These results may provide a crucial clue to recently observed magnetic
structures of the rare-earth-based kagome lattice compounds
$A_2$RE$_3$Sb$_3$O$_{14}$ ($A$ = Mg, Zn; RE = Pr, Nd, Gd, Tb, Dy, Ho, Er, Yb).
