  The equivariant quantum $K$-theory ring of a flag variety is a Frobenius
algebra equipped with a perfect pairing called the quantum $K$-metric. It is
known that in the classical $K$-theory ring for a given flag variety the ideal
sheaf basis is dual to the Schubert basis with regard to the sheaf Euler
characteristic. We define a quantization of the ideal sheaf basis for the
equivariant quantum $K$-theory of cominuscule flag varieties. These quantized
ideal sheaves are then dual to the Schubert basis with regard to the quantum
$K$-metric. We prove explicit type-uniform combinatorial formulae for the
quantized ideal sheaves in terms of the Schubert basis for any cominuscule flag
variety. We also provide an application ultilizing the quantized ideal sheaves
to calculate the Schubert structure constants associated to multiplication by
the top exterior power of the tautological quotient bundle in $QK_T(Gr(k,n))$.
