  The exploration of quantum phases in moir\'e systems has drawn intense
experimental and theoretical efforts. The realization of honeycomb symmetry has
been a recent focus. The combination of strong interaction and honeycomb
symmetry can lead to exotic electronic states such as fractional Chern
insulator, unconventional superconductor, and quantum spin liquid. Accurate
computations in such systems, with reliable treatment of strong long-ranged
Coulomb interaction and approaching the large system sizes to extract
thermodynamic phases, are mostly missing. We study the two-dimensional electron
gas on a honeycomb moir\'e lattice at quarter filling, using fixed-phase
diffusion Monte Carlo. The ground state phases of this important model are
determined in the parameter regime relevant to current experiments. With
increasing moir\'e potential, the systems transitions from a paramagnetic metal
to an itinerant ferromagnetic semimetal and then a charge-density-wave
insulator.
