Markov chain Monte Carlo (MCMC) is now a standard tool for statistical inference, in both Bayesian and frequentist settings. In this talk, I will discuss variance reduction methods for MCMC sampling, including control variates and Rao-Blackwellization. Existing literature in the reversible Markov chain setting will be discussed, and a control variate method will be proposed for reducing the variance and increasing the efficiency of MCMC simulations involving deterministic sweep Markov chains. We establish theoretically that this new control variate approach outperforms competing methodology in several Gibbs sampling settings and further demonstrate the validity of our results via a simulation study. Some future directions and applications for work in this area will be discussed.