  We study the ground state and low-lying excited states of the
Kitaev-Heisenberg model on a ladder geometry using the density matrix
renormalization group and Lanczos exact diagonalization methods. The Kitaev and
Heisenberg interactions are parametrized as $K=\sin\phi$ and $J=\cos\phi$ with
an angle parameter $\phi$. Based on the results for several types of order
parameters, excitation gaps, and entanglement spectra, the $\phi$-dependent
ground-state phase diagram is determined. Remarkably, the phase diagram is
quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice,
exhibiting the same long-range ordered states, namely rung-singlet (analog to
N\'eel in 3D), zigzag, ferromagnetic, and stripy; and the presence of Kitaev
spin liquids around the exactly solvable Kitaev points $\phi=\pm\pi/2$. We also
calculate the expectation value of a plaquette operator corresponding to a
$\pi$-flux state in order to establish how the Kitaev spin liquid extends away
from the $\phi=\pm\pi/2$. Furthermore, we determine the dynamical spin
structure factor and discuss the effect of the Kitaev interaction on the
spin-triplet dispersion.
