  Most research regarding quantum adiabatic optimization has focused on
stoquastic Hamiltonians, whose ground states can be expressed with only real,
nonnegative amplitudes. This raises the question of whether classical Monte
Carlo algorithms can efficiently simulate quantum adiabatic optimization with
stoquastic Hamiltonians. Recent results have given counterexamples in which
path integral and diffusion Monte Carlo fail to do so. However, most adiabatic
optimization algorithms, such as for solving MAX-k-SAT problems, use k-local
Hamiltonians, whereas our previous counterexample for diffusion Monte Carlo
involved n-body interactions. Here we present a new 6-local counterexample
which demonstrates that even for these local Hamiltonians there are cases where
diffusion Monte Carlo cannot efficiently simulate quantum adiabatic
optimization. Furthermore, we perform empirical testing of diffusion Monte
Carlo on a standard well-studied class of permutation-symmetric tunneling
problems and similarly find large advantages for quantum optimization over
diffusion Monte Carlo.
