  Pulsars have been recognized as normal neutron stars, but sometimes argued as
quark stars. {\it Sub-millisecond pulsars, if detected, would play an essential
and important role in distinguishing quark stars from neutron stars.} We focus
on the formation of such sub-millisecond pulsars in this paper. A new approach
to form a sub-millisecond pulsar (quark star) via accretion induced collapse
(AIC) of a white dwarf is investigated here. Under this AIC process, we found
that: (1) almost all the newborn quark stars could have an initial spin period
of $\sim 0.1$ ms; (2) the nascent quark stars (even with a low mass) have
sufficiently high spin-down luminosity and satisfy the conditions for pair
production and sparking process to be as sub-millisecond radio pulsars; (3) in
most cases, the timescales of newborn quark stars in the phase of spin period
$< 1$ (or $<0.5$) ms can be long enough to be detected.
  As a comparison, an accretion spin-up process (for both neutron and quark
stars) is also investigated. It is found that, quark stars formed through AIC
process can have shorter periods ($\leq$ 0.5 ms); while the periods of neutron
stars formed in accretion spin-up process must be longer than 0.5ms. Thus if a
pulsar with a period less than 0.5 ms can be identified in the future, it
should be a quark star.
