We perform first principles calculations of three ordered high-pressure polymorphs of silica to investigate their phase equilibrium: stishovite, CaCl2 and α-PbO2. We use the density-functional linear response theory to calculate lattice dynamical properties and the quasiharmonic approximation to determine the high temperature phase boundaries. We obtain positive and almost parallel Clapeyron slopes for both the stishovite-CaCl2 and the CaCl2-α-PbO2 phase transitions: 56 + 0.006 T (GPa) and 106 + 0.006 T (GPa), respectively. These boundaries are consistent with recent experimental results. The pressure and temperature conditions for the former transition correspond to the mid-lower mantle region where some minor seismic discontinuities have been observed. The latter transition corresponds to the pressure and temperature conditions associated with the lowermost mantle and the D″ discontinuity. The stable form of silica at the bottom of mantle is the α-PbO2 structure.