  In a recent paper [Phys. Rev. D 92:084012, 2015], the author studied the
classical $(1+1)$-dimensional Friedmann-Robertson-Walker (FRW) universe filled
with a perfect fluid in Ho\v{r}ava-Lifshitz (HL) theory of gravity. This theory
is dynamical due to the anisotropic scaling of space and time. It also
resembles the Jackiw-Teitelboim model, in which a dilatonic degree of freedom
is necessary for dynamics. In this paper, I will give one step further in the
understanding of (1+1)-dimensional HL cosmology by means of the quantization of
the FRW universe filled with a perfect fluid with equation of state (EoS)
$p=w\rho$. The fluid will be introduced in the model via Schutz formalism and
Dirac's algorithm will be used for quantization. It will be shown that the
Schr\"odinger equation for the wave function of the universe has the following
properties: for $w=1$ (radiation fluid), the characteristic potential will be
exponential, resembling Liouville quantum mechanics; for $w\neq 1$, a
characteristic inverse square potential appears in addition to a regular
polynomial which depends on the EoS. Explict solutions for a few cases of
interesting will be found and the expectation value of the scale factor will be
calculated. As in usual quantum cosmology, it will be shown that the quantum
theory smooth out the big-bang singularity, but the classical behavior of the
universe is recovered in the low-energy limit.
