A recent claim from the LHCb collaboration on the observation of two hidden-charm pentaquark states revives hopes for experimental discoveries of other multiquark baryonic states such as the H-dibaryon with a 6-quark ($uuddss$) configuration. Recent theoretical predictions for the mass of H-dibaryon pointing to the mass region near the $\Lambda\Lambda$ threshold also encourage experimental searches. We have proposed a dedicated experiment to search for the H-dibaryon in the bound and unbound mass regions near $\Lambda\Lambda$ threshold. We plan to measure production of $\Lambda p\pi^-$, $\Lambda\Lambda$ and $\Xi^-p$ systems in the $^{12}$C$(K^-,K^+)$ reaction at J-PARC. We have constructed a large-acceptance hyperon spectrometer consisting of a superconducting dipole magnet and a time projection chamber, which allows conclusive experimental searches with a precise 1-MeV mass resolution.