Staggering computational and algorithmic advances in recent years now make possible systematic Quantum Monte Carlo (QMC) simulations of high temperature (high-Tc) superconductivity in a microscopic model, the two dimensional (2D) Hubbard model, with parameters relevant to the cuprate materials. Here we report the algorithmic and computational advances that enable us to study the effect of disorder and nano-scale inhomogeneities on the pair-formation and the superconducting transition temperature necessary to understand real materials. The simulation code is written with a generic and extensible approach and is tuned to perform well at scale. Significant algorithmic improvements have been made to make effective use of current supercomputing architectures. By implementing delayed Monte Carlo updates and a mixed single-/double precision mode, we are able to dramatically increase the efficiency of the code. On the Cray XT4 systems of the Oak Ridge National Laboratory (ORNL), for example, we currently run production jobs on 31 thousand processors and thereby routinely achieve a sustained performance that exceeds 200 TFlop/s. On a system with 49 thousand processors we achieved a sustained performance of 409 TFlop/s. We present a study of how random disorder in the effective Coulomb interaction strength affects the superconducting transition temperature in the Hubbard model.