We present a new class of high-order imaginary time propagators for path integral Monte Carlo simulations that require no higher order derivatives of the potential nor explicit quadratures of Gaussian trajectories. Higher orders are achieved by an extrapolation of the primitive second-order propagator involving subtractions. By requiring all terms of the extrapolated propagator to have the same Gaussian trajectory, the subtraction only affects the potential part of the path integral. The resulting violation of positivity has surprisingly little effects on the accuracy of the algorithms at practical time steps. Thus in principle, arbitrarily high order algorithms can be devised for path integral Monte Carlo simulations. We verified the fourth, sixth, and eighth order convergences of these algorithms by solving for the ground state energy and pair distribution function of liquid (4)He, which is representative of a dense, and strongly interacting, quantum many-body system.