  A quantum $n$-particle model consisting of an open $q$-difference Toda chain
with two-sided boundary interactions is placed on a finite integer lattice. The
spectrum and eigenbasis are computed by establishing the equivalence with a
previously studied $q$-boson model from which the quantum integrability is
inherited. Specifically, the $q$-boson-Toda correspondence in question yields
Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall-Littlewood
polynomials and provides the pertinent solutions of the Bethe Ansatz equations
via the global minima of corresponding Yang-Yang type Morse functions.
