  Understanding of the role of X-rays for driving the thermal evolution of the
intergalactic medium (IGM) at high redshifts is one of important questions in
astrophysics. High-mass X-ray binaries (HMXBs) in early stellar populations are
prime X-ray source; however, their formation efficiency is not well understood.
Using $N$-body simulations, we estimate the HMXB formation rate via mutual
gravitational interactions of nascent, small groups of the Population~III
stars. We find that HMXBs form at a rate of one per $\gtrsim 10^{4}M_{\odot}$
in newly born stars, and that they emit with a power of $\sim 10^{41} {\rm
erg}~{\rm s}^{-1}$ in the $2-10$ keV band per star formation rate (SFR). This
value is a factor $\sim 10^{2}$ larger than what is observed in star forming
galaxies at lower redshifts; the X-ray production from early HMXBs would have
been even more copious, if they also formed \textit{in situ} or via migration
in protostellar disks. Combining our results with earlier studies suggests that
early HMXBs were highly effective at heating the IGM and leaving a strong 21 cm
signature. We discuss broader implications of our results, such as the rate of
long gamma-ray bursts from Population~III stars and the direct collapse channel
for massive black hole formation.
