  Superconductivity in doped quantum paramagnets has been a subject of long
theoretical inquiry. In this work we report a density matrix renormalization
group study of lightly doped $t$-$J$ models on the square lattice (doped hole
densities $\delta = 1/12$ and 1/8) with parameters for which previous studies
have suggested that the undoped system in 2D is either a quantum spin liquid or
a valence bond crystal. Our studies are performed on cylinders with width up to
8. Ground-state correlations are found to be nearly identical for the ``doped
quantum spin liquid'' and ``doped valence bond crystal''. Upon increasing the
cylinder width from 4 to 8, we observed a significant strengthening of the
quasi-long-range superconducting correlations, and a dramatic suppression of
any ``competing'' charge-density-wave order. Extrapolating from the observed
behavior of the width 8 cylinders, we speculate that the system has a nodeless
d-wave superconducting ground-state in the 2D limit.
