  The graded off-diagonal Bethe ansatz method is proposed to study
supersymmetric quantum integrable models (i.e., quantum integrable models
associated with superalgebras). As an example, the exact solutions of the
$SU(2|2)$ vertex model with both periodic and generic open boundary conditions
are constructed. By generalizing the fusion techniques to the supersymmetric
case, a closed set of operator product identities about the transfer matrices
are derived, which allows us to give the eigenvalues in terms of homogeneous or
inhomogeneous $T-Q$ relations. The method and results provided in this paper
can be generalized to other high rank supersymmetric quantum integrable models.
