  We compute the planar finite size corrections to the spectrum of the
dilatation operator acting on two-impurity states of a certain limit of
conformal $\mathcal{N}=2$ quiver gauge field theory which is a $Z_M$-orbifold
of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We match the result to the
string dual, IIB superstrings propagating on a pp-wave background with a
periodically identified null coordinate. Up to two loops, we show that the
computation of operator dimensions, using an effective Hamiltonian technique
derived from renormalized perturbation theory and a twisted Bethe ansatz which
is a simple generalization of the
Beisert-Dippel-Staudacher~\cite{Beisert:2004hm} long range spin chain, agree
with each other and also agree with a computation of the analogous quantity in
the string theory. We compute the spectrum at three loop order using the
twisted Bethe ansatz and find a disagreement with the string spectrum very
similar to the known one in the near BMN limit of $\mathcal{N}=4$
super-Yang-Mills theory. We show that, like in $\mathcal{N}=4$, this
disagreement can be resolved by adding a conjectured ``dressing factor'' to the
twisted Bethe ansatz. Our results are consistent with integrability of the
$\mathcal{N}=2$ theory within the same framework as that of $\mathcal{N}=4$.
