High pressure systems are important, for example, to understand the interiors of giant planets (Jupiter and Saturn), for experiments at NIF (the National Ignition Facility at Livermore) related to inertially confined fusion and for other interests of DOE. In this project, we are developing innovative simulation methods (Quantum Monte Carlo methods) to allow more accurate calculation of properties of systems under extreme conditions of pressure and temperature. These methods can use the power of current day supercomputers made of very many processors, starting from the basic equations of physics to model quantum phenomena important at the microscopic scale. During the grant period, we have settled two important questions of the physics of hydrogen and helium under extreme conditions. We have found the pressures and temperatures when hydrogen and helium mix together; this is important to understand the difference of the interiors of the planets Jupiter and Saturn. Secondly, we have shown that there exists a sharp transition as a function of pressure between molecular and atomic liquid hydrogen at temperatures below 2000K. This prediction can be confirmed with high pressure experiments.