  We report the first results of a high-redshift ($z$ >~ 5) quasar survey using
the Dark Energy Spectroscopic Instrument (DESI). As a DESI secondary target
program, this survey is designed to carry out a systematic search and
investigation of quasars at $z$ >~ 5, up to redshift 6.8. The target selection
is based on the DESI Legacy Imaging Surveys (the Legacy Surveys) DR9
photometry, combined with the Pan-STARRS1 data and $J$-band photometry from
public surveys. A first quasar sample has been constructed from the DESI Survey
Validation 3 (SV3) and first-year observations until May 2022. This sample
includes more than 400 new quasars at redshift 4.7 <= $z$ < 6.6, down to 21.5
magnitude in the $z$ band, discovered from 35% of the entire target sample.
Remarkably, there are 220 new quasars identified at $z$ >= 5, more than one
third of existing quasars previously published at this redshift. The
observations so far result in an average success rate of 23% at $z$ > 4.7. The
current spectral dataset has already allowed analysis of interesting individual
objects (e.g., quasars with damped Ly$\alpha$ absorbers and broad absorption
line features), and statistical analysis will follow the survey's completion. A
set of science projects will be carried out leveraging this program, including
quasar luminosity function, quasar clustering, intergalactic medium, quasar
spectral properties, intervening absorbers, and properties of early
supermassive black holes. Additionally, a sample of 38 new quasars at $z$ ~
3.8-5.7 discovered from a pilot survey in the DESI SV1 is also published in
this paper.
