Modeling large-scale sky survey observations is a key driver for the continuing development of high-resolution, large-volume, cosmological simulations. We report the first results from the" Q Continuum" cosmological N-body simulation run carried out on the GPU-accelerated supercomputer Titan. The simulation encompasses a volume of and evolves more than half a trillion particles, leading to a particle mass resolution of . At this mass resolution, the Q Continuum run is currently the largest cosmology simulation available. It enables the construction of detailed synthetic sky catalogs, encompassing different modeling methodologies, including semi-analytic modeling and sub-halo abundance matching in a large, cosmological volume. Here we describe the simulation and outputs in detail and present first results for a range of cosmological statistics, such as mass power spectra, halo mass functions, and halo mass-concentration relations for different epochs. We also provide details on challenges connected to running a simulation on almost 90% of Titan, one of the fastest supercomputers in the world, including our usage of Titan's GPU accelerators.