  The Magn\'eli phase Ti$_4$O$_7$ is an important transition metal oxide with a
wide range of applications because of its interplay between charge, spin, and
lattice degrees of freedom. At low temperatures, it has non-trivial magnetic
states very close in energy, driven by electronic exchange and correlation
interactions. We have examined three low-lying states, one ferromagnetic and
two antiferromagnetic, and calculated their energies as well as Ti spin moment
distributions using highly accurate Quantum Monte Carlo methods. We compare our
results to those obtained from density functional theory-based methods that
include approximate corrections for exchange and correlation. Our results
confirm the nature of the states and their ordering in energy, as compared with
density-functional theory methods. However, the energy differences and spin
distributions differ. A detailed analysis suggests that non-local
exchange-correlation functionals, in addition to other approximations such as
LDA+U to account for correlations, are needed to simultaneously obtain better
estimates for spin moments, distributions, energy differences and energy gaps.
