  Centers of starburst galaxies may be characterized by a specific gas and ice
chemistry due to their gas dynamics and the presence of various ice desorption
mechanisms. This may result in a peculiar observable composition. We analyze
abundances of $CO_2$, a reliable tracer of ice chemistry, from data collected
as part of the ALMA large program ALCHEMI, a wide-frequency spectral scan
toward the starburst galaxy NGC~253 with an angular resolution of 1.6$''$. We
constrain the $CO_2$ abundances in the gas phase using its protonated form
$HOCO^+$. The distribution of $HOCO^+$ is similar to that of methanol, which
suggests that $HOCO^+$ is indeed produced from the protonation of $CO_2$
sublimated from ice. The $HOCO^+$ fractional abundances are found to be
$(1-2)\times10^{-9}$ at the outer part of the central molecular zone (CMZ),
while they are lower ($\sim10^{-10}$) near the kinematic center. This peak
fractional abundance at the outer CMZ is comparable to that in the Milky Way
CMZ, and orders of magnitude higher than that in Galactic disk star-forming
regions. From the range of $HOCO^+/CO_2$ ratios suggested from chemical models,
the gas-phase $CO_2$ fractional abundance is estimated to be
$(1-20)\times10^{-7}$ at the outer CMZ, and orders of magnitude lower near the
center. We estimate the $CO_2$ ice fractional abundances at the outer CMZ to be
$(2-5)\times10^{-6}$ from the literature. A comparison between the ice and gas
$CO_2$ abundances suggests an efficient sublimation mechanism. This sublimation
is attributed to large-scale shocks at the orbital intersections of the bar and
CMZ.
