  For each positive rational number $\epsilon$, we define $K$-theoretic
$\epsilon$-stable quasimaps to certain GIT quotients $W\sslash G$. For
$\epsilon>1$, this recovers the $K$-theoretic Gromov-Witten theory of $W\sslash
G$ introduced in more general context by Givental and Y.-P. Lee.
  For arbitrary $\epsilon_1$ and $\epsilon_2$ in different stability chambers,
these $K$-theoretic quasimap invariants are expected to be related by
wall-crossing formulas. We prove wall-crossing formulas for genus zero
$K$-theoretic quasimap theory when the target $W\sslash G$ admits a torus
action with isolated fixed points and isolated one-dimensional orbits.
