  We carry out a feasibility study toward a lattice QCD calculation of the
neutron electric dipole moment (NEDM) in the presence of the $\theta$ term
using two different approaches. In the first method, we calculate the CP-odd
electromagnetic form factor $F_3$, which becomes the NEDM in the zero momentum
transfer limit. At the first order in $\theta$, we derive a formula connecting
the lattice three-point function to the CP-odd electromagnetic form factor. In
the second method we directly extract the NEDM from the energy difference
between spin-up and spin-down neutron states in the presence of a constant
electric field, without expanding a small but non-zero $\theta$. We test both
approaches numerically, employing the domain-wall quark action with the RG
improved gauge action in quenched QCD at $a^{-1}\simeq 2$ GeV on a $16^3\times
32\times 16$ lattice, and further applying the second method to the clover
quark action at a similar lattice spacing and nucleon mass. We obtain good
signals from both approaches. In particular the second method works well with
both fermion formulations.
