Cosmology places the strongest current limits on the sum of neutrino masses. Future observations will further improve the sensitivity and this will require accurate cosmological simulations to quantify possible systematic uncertainties and to make predictions for non-linear scales, where much information resides. However, shot noise arising from neutrino thermal motions limits the accuracy of simulations. In this paper, we introduce a new method for simulating large-scale structure formation with neutrinos that accurately resolves the neutrinos down to small scales and significantly reduces the shot noise. The method works by tracking perturbations to the neutrino phase-space distribution with particles and reduces shot noise in the power spectrum by a factor of O (102) at z = 0 for minimal neutrino masses and significantly more at higher redshifts, without neglecting the back-reaction caused by neutrino clustering. We prove that the method is part of a family of optimal methods that minimize shot noise subject to a maximum deviation from the non-linear solution. Compared to other methods, we find per mille level agreement in the matter power spectrum and per cent level agreement in the large-scale neutrino bias, but large differences in the neutrino component on small scales. A basic version of the method can easily be implemented in existing N-body codes and allows neutrino simulations with significantly reduced particle load. Further gains are possible by constructing background models based on perturbation theory. A major advantage of this technique is that it works well for all masses, enabling a consistent exploration of the full neutrino parameter space
