  In this paper, we investigate the HAL QCD potential in the $I=1$ $\pi \pi$
scattering using the hybrid method for all-to-all propagators, in which a
propagator is approximated by low-eigenmodes and the remaining high-eigenmode
part is stochastically estimated. To verify the applicability of the hybrid
method to systems containing quark creation$/$annihilation contributions such
as the $\rho$ meson, we calculate the $I=1$ $\pi\pi$ potential with the 2+1
flavor gauge configurations on $16^3 \times 32$ lattice at the lattice spacing
$a \approx 0.12$ fm and $(m_{\pi},m_{\rho}) \approx (870, 1230)$ MeV, in which
the $\rho$ meson appears as a deeply-bound state. While we find that the naive
stochastic evaluations for quark creation$/$annihilation contributions lead to
extremely large statistical fluctuations, additional noise reduction methods
enable us to obtain a sufficiently precise potential, which shows a strong
attractive force. We also confirm that the binding energy and $k^3 \cot \delta$
obtained from our potential are roughly consistent with an existing $\rho$
meson bound state, within a large systematic error associated with our
calculation, whose possible origin is also discussed.
