  Approximate Bayesian Computation (ABC) is a method to obtain a posterior
distribution without a likelihood function, using simulations and a set of
distance metrics. For that reason, it has recently been gaining popularity as
an analysis tool in cosmology and astrophysics. Its drawback, however, is a
slow convergence rate. We propose a novel method, which we call qABC, to
accelerate ABC with Quantile Regression. In this method, we create a model of
quantiles of distance measure as a function of input parameters. This model is
trained on a small number of simulations and estimates which regions of the
prior space are likely to be accepted into the posterior. Other regions are
then immediately rejected. This procedure is then repeated as more simulations
are available. We apply it to the practical problem of estimation of redshift
distribution of cosmological samples, using forward modelling developed in
previous work. The qABC method converges to nearly same posterior as the basic
ABC. It uses, however, only 20\% of the number of simulations compared to basic
ABC, achieving a fivefold gain in execution time for our problem. For other
problems the acceleration rate may vary; it depends on how close the prior is
to the final posterior. We discuss possible improvements and extensions to this
method.
