  Titanium dioxide, TiO$_2$, has multiple applications in catalysis, energy
conversion and memristive devices because of its electronic structure. Most of
these applications utilize the naturally existing phases: rutile, anatase and
brookite. Despite the simple form of TiO$_2$ and its wide uses, there is
long-standing disagreement between theory and experiment on the energetic
ordering of these phases that has never been resolved. We present the first
analysis of phase stability at zero temperature using the highly accurate
many-body fixed node diffusion Quantum Monte Carlo (QMC) method. We also
include the effects of temperature by calculating the Helmholtz free energy
including both internal energy and vibrational contributions from density
functional perturbation theory based quasi harmonic phonon calculations. Our
QMC calculations find that anatase is the most stable phase at zero
temperature, consistent with many previous mean-field calculations. However, at
elevated temperatures, rutile becomes the most stable phase. For all finite
temperatures, brookite is always the least stable phase.
