  The rational $Q$-system is an efficient method to solve Bethe ansatz
equations for quantum integrable spin chains. We construct the rational
$Q$-systems for generic Bethe ansatz equations described by an $A_{\ell-1}$
quiver, which include models with multiple momentum carrying nodes, generic
inhomogeneities, generic diagonal twists and $q$-deformation. The rational
$Q$-system thus constructed is specified by two partitions. Under Bethe/Gauge
correspondence, the rational $Q$-system is in a one-to-one correspondence with
a 3d $\mathcal{N}=4$ quiver gauge theory of the type
${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$, which is also specified
by the same partitions. This shows that the rational $Q$-system is a natural
language for the Bethe/Gauge correspondence, because known features of the
${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$ theories readily
translate. For instance, we show that the Higgs and Coulomb branch Higgsing
correspond to modifying one of the partitions in the rational $Q$-system while
keeping the other untouched. Similarly, mirror symmetry is realized in terms of
the rational $Q$-system by simply swapping the two partitions - exactly as for
${T}_{\boldsymbol{\rho}}^{\boldsymbol{\sigma}}[SU(n)]$. We exemplify the
computational efficiency of the rational $Q$-system by evaluating topologically
twisted indices for 3d $\mathcal{N}=4$ $U(n)$ SQCD theories with
$n=1,\ldots,5$.
