  In Ref [1] we have presented the results of an exploratory lattice QCD
computation of the long-distance contribution to the $K^+\to\pi^+\nu\bar{\nu}$
decay amplitude. In the present paper we describe the details of this
calculation, which includes the implementation of a number of novel techniques.
The $K^+\to\pi^+\nu\bar{\nu}$ decay amplitude is dominated by short-distance
contributions which can be computed in perturbation theory with the only
required non-perturbative input being the relatively well-known form factors of
semileptonic kaon decays. The long-distance contributions, which are the target
of this work, are expected to be of O(5%) in the branching ratio. Our study
demonstrates the feasibility of lattice QCD computations of the
$K^+\to\pi^+\nu\bar{\nu}$ decay amplitude, and in particular of the
long-distance component. Though this calculation is performed on a small
lattice ($16^3\times32$) and at unphysical pion, kaon and charm quark masses,
$m_\pi=420$ MeV, $m_K=563$ MeV and $m_c^{\overline{\mathrm{MS}}}(\mbox{2
GeV})=863$ MeV, the techniques presented in this work can readily be applied to
a future realistic calculation.
