  Markov chains can be used to generate samples whose distribution approximates
a given target distribution. The quality of the samples of such Markov chains
can be measured by the discrepancy between the empirical distribution of the
samples and the target distribution. We prove upper bounds on this discrepancy
under the assumption that the Markov chain is uniformly ergodic and the driver
sequence is deterministic rather than independent $U(0,1)$ random variables. In
particular, we show the existence of driver sequences for which the discrepancy
of the Markov chain from the target distribution with respect to certain test
sets converges with (almost) the usual Monte Carlo rate of $n^{-1/2}$.
