  The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by
Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf
subring of the ring of symmetric functions. On the other hand, for the quantum
$K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a
conjectural presentation based on the results obtained by Givental and Lee. We
construct an explicit birational morphism between the spectrums of these two
rings. Our method relies on Ruijsenaars's relativistic Toda lattice with
unipotent initial condition. From this result, we obtain a $K$-theory analogue
of the so-called Peterson isomorphism for (co)homology. We provide a conjecture
on the detailed relationship between the Schubert bases, and, in particular, we
determine the image of Lenart--Maeno's quantum Grothendieck polynomial
associated with a Grassmannian permutation.
