Nuclear matter is an ideal testbed for nuclear interactions with important consequences for finite nuclei as well as nuclear astrophysics. While recent *ab initio* calculations of medium-mass to heavy nuclei have demonstrated the importance of realistic saturation properties of infinite matter for nuclear forces derived in chiral effective field theory, the equation of state allows tight constraints, e.g., on the symmetry energy.

We present an efficient Monte-Carlo framework for perturbative calculations of infinite nuclear matter based on chiral two-, three-, and four-nucleon interactions, which enables the incorporation of all many-body contributions in a straightforward and transparent way. Furthermore, it makes it possible to extract systematic uncertainty estimates by performing order-by-order calculations in the chiral as well as the many-body expansion. In a first application, we make use of this novel framework to explore new chiral interactions up to next-to-next-to-next-to-leading order (N^{3}LO). Remarkably, simultaneous fits to the triton and to saturation properties can be achieved with natural 3N low-energy couplings.

Finally, we present a comprehensive Weinberg eigenvalue analysis of a representative set of modern chiral nucleon-nucleon interactions. Our results provide insights into the perturbativeness and scheme dependencies of these local, semilocal, and nonlocal potentials.