  We investigate a quantum phase transition between a Neel phase and a valence
bond solid (VBS) phase, in each of which a different Z2 symmetry is broken, in
a spin-1/2 two-leg XXZ ladder with a four-spin interaction. The model can be
viewed as a one-dimensional variant of the celebrated J-Q model on a square
lattice. By means of variational uniform matrix product state calculations and
an effective field theory, we determine the phase diagram of the model, and
present evidences that the Neel-VBS transition is continuous and belongs to the
Gaussian universality class with the central charge c=1. In particular, the
critical exponents $\beta, \eta,$ and, $\nu$ are found to satisfy the
constraints expected for a Gaussian transition within numerical accuracy. These
exponents do not detectably change along the phase boundary while they are in
general allowed to do so for the Gaussian class.
