Equation of states for deuterium, helium, and their mixture is studied by using the quantum molecular dynamics (QMD) method. We calculate the equation of states for helium with density from 0.32 to 5 g/cm3 at temperature from 1000 to 50000 K. Results are compared with the chemical model (CM), at T less than 10000 K, and QMD is in good agreement with the CM. The shock Hugoniot curves are also calculated, and the results are in good agreement with the gas-gun experiment. The mechanism of the metal-insulator transition for helium is studied by computing its pair distribution function and density of states. The equation of states (EOS) for deuterium with density from 0.19 to 0.84 g/cm3 at temperatures from 20 to 50000 K is computed. For deuterium molecule the degree of dissociation is calculated, and the effect of the molecular vibration is accounted for using the EOS model. Theoretical Hugoniot states are also calculated and compared with the results of experiments and other theories; the maximum compressibility of hydrogen is about 4.9, and deuterium 4.4; these agree with the results of most experiments and theories. Due to the zero point motion of atoms being not taken into account, the theoretical results at low temperatures are smaller than those of experiments. The deuterium-helium mixture is studied, and its 293 points of equation of states for various xHe with densities from 0.19 to 0.84 g/cm3 at temperatures from 100 to 50000 K are calculated. The linear mixing approximation (LMA) is checked, and the maximum of the volume distinction is about 7%; the results indicate that LMA is a cursory approximation.