  We present the first quantum Monte Carlo (QMC) calculations with chiral
effective field theory (EFT) interactions. To achieve this, we remove all
sources of nonlocality, which hamper the inclusion in QMC calculations, in
nuclear forces to next-to-next-to-leading order. We perform auxiliary-field
diffusion Monte Carlo (AFDMC) calculations for the neutron matter energy up to
saturation density based on local leading-order, next-to-leading order, and
next-to-next-to-leading order nucleon-nucleon interactions. Our results exhibit
a systematic order-by-order convergence in chiral EFT and provide
nonperturbative benchmarks with theoretical uncertainties. For the softer
interactions, perturbative calculations are in excellent agreement with the
AFDMC results. This work paves the way for QMC calculations with systematic
chiral EFT interactions for nuclei and nuclear matter, for testing the
perturbativeness of different orders, and allows for matching to lattice QCD
results by varying the pion mass.
