  The s-wave pion-kaon ($\pi K$) scattering lengths at zero momentum are
calculated in lattice QCD with sufficiently light $u/d$ quarks and strange
quark at its physical value by the finite size formula. The light quark masses
correspond to $m_\pi = 0.330 - 0.466$ GeV. In the "Asqtad" improved staggered
fermion formulation, we measure the $\pi K$ four-point correlators for both
isospin $I=1/2$ and 3/2 channels, and analyze the lattice simulation data at
the next-to-leading order in the continuum three-flavor chiral perturbation
theory, which enables us a simultaneous extrapolation of $\pi K$ scattering
lengths at physical point. We adopt a technique with the moving wall sources
without gauge fixing to obtain the substantiable accuracy, moreover, for $I =
1/2$ channel, we employ the variational method to isolate the contamination
from the excited states. Extrapolating to the physical point yields the
scattering lengths as $m_\pi a_{3/2} = -0.0505(19)$ and $m_\pi a_{1/2} =
0.1827(37)$ for $I=3/2$ and 1/2 channels, respectively. Our simulation results
for $\pi K$ scattering lengths are in agreement with the experimental reports
and theoretical predictions, and can be comparable with other lattice
simulations. These simulations are carried out with MILC $N_f = 2+1$ flavor
gauge configurations at lattice spacing $a \approx 0.15$ fm.
