  We prove that any unitary highest weight module over a universal minimal
quantum affine $W$-algebra at non-critical level descends to its simple
quotient. We find the defining relations of the unitary simple minimal quantum
affine $W$-algebras and the list of all their irreducible positive energy
modules. We also classify all irreducible highest weight modules for the simple
affine vertex algebras in the cases when the associated simple minimal
$W$-algebra is unitary.
