  We have performed quantum Monte Carlo (QMC) simulations and density
functional theory (DFT) calculations to study the equations of state of
MgSiO$_3$ perovskite (Pv) and post-perovskite (PPv), up to the pressure and
temperature conditions of the base of Earth's lower mantle. The ground state
energies were derived using QMC and the temperature dependent Helmholtz free
energies were calculated within the quasi-harmonic approximation and density
functional perturbation theory. The equations of state for both phases of
MgSiO$_3$ agree well with experiments, and better than those from generalized
gradient approximation (GGA) calculations. The Pv-PPv phase boundary calculated
from our QMC equations of states is also consistent with experiments, and
better than previous LDA calculations. We discuss the implications for double
crossing of the Pv-PPv boundary in the Earth.
