  We show that for magic-angle twisted bilayer graphene (TBG) away from charge
neutrality, although quantum Monte Carlo (QMC) simulations suffer from the sign
problem, the computational complexity is at most polynomial at certain integer
fillings. For even integer fillings, this polynomial complexity survives even
if an extra inter-valley attractive interaction is introduced, on top of
Coulomb repulsions. This observation allows us to simulate magic-angle twisted
bilayer graphene and to obtain accurate phase diagram and dynamical properties.
At the chiral limit and filling $\nu=1$, the simulations reveal a thermodynamic
transition separating metallic state and a $C=1$ correlated Chern insulator --
topological Mott insulator (TMI) -- and the pseudogap spectrum slightly above
the transition temperature. The ground state excitation spectra of the TMI
exhibit a spin-valley U(4) Goldstone mode and a time reversal restoring
excitonic gap smaller than the single particle gap. These results are
qualitatively consistent with the recent experimental findings at zero-field
and $\nu=1$ filling in $h$-BN nonaligned TBG.
