  Markov chain Monte Carlo (MCMC) simulations are modeled as driven by true
random numbers. We consider variance bounding Markov chains driven by a
deterministic sequence of numbers. The star-discrepancy provides a measure of
efficiency of such Markov chain quasi-Monte Carlo methods. We define a
pull-back discrepancy of the driver sequence and state a close relation to the
star-discrepancy of the Markov chain-quasi Monte Carlo samples. We prove that
there exists a deterministic driver sequence such that the discrepancies
decrease almost with the Monte Carlo rate $n^{1/2}$. As for MCMC simulations, a
burn-in period can also be taken into account for Markov chain quasi-Monte
Carlo to reduce the influence of the initial state. In particular, our
discrepancy bound leads to an estimate of the error for the computation of
expectations. To illustrate our theory we provide an example for the Metropolis
algorithm based on a ball walk. Furthermore, under additional assumptions we
prove the existence of a driver sequence such that the discrepancy of the
corresponding deterministic Markov chain sample decreases with order
$n^{-1+\delta}$ for every $\delta>0$.
