  Many fundamental results of pluripotential theory on the quaternionic space
$\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of a
plurisubharmonic function, the quaternionic Monge-Amp\`{e}re operator,
differential operators $d_0$ and $d_1$ and a closed positive current on the
Heisenberg group. The quaternionic Monge-Amp\`{e}re operator is the coefficient
of $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the
existence of quaternionic Monge-Amp\`{e}re measure for a continuous
quaternionic plurisubharmonic function and the minimum principle for the
quaternionic Monge-Amp\`{e}re operator. Unlike the tangential Cauchy-Riemann
operator $ \overline{\partial}_b $ on the Heisenberg group which behaves badly
as $ \partial_b\overline{\partial}_b\neq -\overline{\partial}_b\partial_b $,
the quaternionic counterpart $d_0$ and $d_1$ satisfy $ d_0d_1=-d_1d_0 $. This
is the main reason that we have a better theory for the quaternionic
Monge-Amp\`{e}re operator than $ (\partial_b\overline{\partial}_b)^n$.
