We present redshift space two-point ($xi$), three-point ($zeta$) and reduced three-point (Q) correlation of Ly$alpha$ absorbers (i.e Voigt profile components having HI column density $N_{HI}>10^{13.5}$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 $zeta$ is seen in $z=1.7-2.3$ redshift bin at $1-2h^{-1}$cMpc with an amplitude of $1.81pm0.59$ ($sim3.1sigma$ level). The corresponding Q is found to be $0.68pm0.23$. The measured $xi$ and $zeta$ values show an increasing trend with $N_{HI}$, while Q remains relatively independent of $N_{HI}$. We find $xi$ and $zeta$ to evolve strongly with $z$ over the redshift range studied. Using hydrodynamical simulations, we find that the $xi$ and especially $zeta$ seen in real space may be strongly amplified by peculiar velocities in redshift space. Simulations also suggest that while feedback, thermal and pressure smoothing effects affect the clustering of Ly$alpha$ absorbers at small scales, i.e $<0.5h^{-1}$cMpc, the HI photo-ionization rates ($Gamma_{HI}$) strongly influence the correlation amplitudes at all scales. We find that the strong redshift evolution shown by $xi$ and $zeta$ is primarily sourced by the redshift evolution of the relationship between $N_{HI}$ and baryon overdensity ($Delta$). Our simulations that uses available best fitted $Gamma_{HI}(z)$ measurements produce consistent clustering signals with observations at $zsim2$ but under-predict the clustering at higher redshifts. One possible remedy is to have higher values of $Gamma_{HI}$ at higher redshifts compared to the existing measurements. Alternatively the discrepancy could be related to non-equilibrium and inhomogeneous conditions prevailing during HeII reionization not captured by our simulations.
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