  We present redshift space two-point ($\xi$), three-point ($\zeta$) and
reduced three-point (Q) correlations of Ly$\alpha$ absorbers (Voigt profile
components having HI column density, $N_{\rm HI}>10^{13.5}$cm$^{-2}$) over
three redshift bins spanning $1.7< z<3.5$ using high-resolution spectra of 292
quasars. We detect positive $\xi$ up to 8 $h^{-1}$ cMpc in all three redshift
bins. The strongest detection of $\zeta =1.81\pm 0.59$ (with Q$=0.68\pm 0.23$),
is in $z=1.7-2.3$ bin at $1-2h^{-1}$ cMpc. The measured $\xi$ and $\zeta$
values show an increasing trend with $N_{\rm HI}$, while Q remains relatively
independent of $N_{\rm HI}$. We find $\xi$ and $\zeta$ to evolve strongly with
redshift. Using simulations, we find that $\xi$ and $\zeta$ seen in real space
may be strongly amplified by peculiar velocities in redshift space. Simulations
suggest that while feedback, thermal and pressure smoothing effects influence
the clustering of Ly$\alpha$ absorbers at small scales, i.e $<0.5h^{-1}$ cMpc,
the HI photo-ionization rate ($\Gamma_{\rm HI}$) has a strong influence at all
scales. The strong redshift evolution of $\xi$ and $\zeta$ (for a fixed $N_{\rm
HI}$-cutoff) is driven by the redshift evolution of the relationship between
$N_{\rm HI}$ and baryon overdensity. Our simulation using best-fitted
$\Gamma_{\rm HI}(z)$ measurements produces consistent clustering signals with
observations at $z\sim 2$ but under-predicts the clustering at higher
redshifts. One possible remedy is to have higher values of $\Gamma_{\rm HI}$ at
higher redshifts. Alternatively the discrepancy could be related to
non-equilibrium and inhomogeneous conditions prevailing during HeII
reionization not captured by our simulations.
