We present a sample of 38 intervening Damped Lyman $alpha$ (DLA) systems identified towards 100 $z>3.5$ quasars, observed during the XQ-100 survey. The XQ-100 DLA sample is combined with major DLA surveys in the literature. The final combined sample consists of 742 DLAs over a redshift range approximately $1.6 < z_{rm abs} < 5.0$. We develop a novel technique for computing $Omega_{rm HI}^{rm DLA}$ as a continuous function of redshift, and we thoroughly assess and quantify the sources of error therein, including fitting errors and incomplete sampling of the high column density end of the column density distribution function. There is a statistically significant redshift evolution in $Omega_{rm HI}^{rm DLA}$ ($geq 3 sigma$) from $z sim 2$ to $z sim$ 5. In order to make a complete assessment of the redshift evolution of $Omega_{rm HI}$, we combine our high redshift DLA sample with absorption surveys at intermediate redshift and 21cm emission line surveys of the local universe. Although $Omega_{rm HI}^{rm DLA}$, and hence its redshift evolution, remains uncertain in the intermediate redshift regime ($0.1 < z_{rm abs} < 1.6$), we find that the combination of high redshift data with 21cm surveys of the local universe all yield a statistically significant evolution in $Omega_{rm HI}$ from $z sim 0$ to $z sim 5$ ($geq 3 sigma$). Despite its statistical significance, the magnitude of the evolution is small: a linear regression fit between $Omega_{rm HI}$ and $z$ yields a typical slope of $sim$0.17$times 10^{-3}$, corresponding to a factor of $sim$ 4 decrease in $Omega_{rm HI}$ between $z=5$ and $z=0$.
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