We report new Chandra observations of seven optically faint, z \sim 4 radio-quiet quasars. We have combined these new observations with previous Chandra observations of radio-quiet quasars to create a sample of 174 sources. These sources have 0.1 < z < 4.7, and 10^{44} ergs s^{-1} < nu L_{nu} (2500 Å) < 10^{48} ergs s^{-1}. The X-ray detection fraction is 90%. We find that the X-ray loudness of radio-quiet quasars decreases with UV luminosity and increases with redshift. The model that is best supported by the data has a linear dependence of optical-to-X-ray ratio, alpha_{ox}, on cosmic time, and a quadratic dependence of alpha_{ox} on log L_{UV}, where alpha_{ox} becomes X-ray quiet more rapidly at higher log L_{UV}. We find no significant evidence for a relationship between the X-ray photon index, Gamma_X, and the UV luminosity, and we find marginally significant evidence that the X-ray continuum flattens with increasing z (2 sigma). The Gamma_X-z anti-correlation may be the result of X-ray spectral curvature, redshifting of a Compton reflection component into the observed Chandra band, and/or redshifting of a soft excess out of the observed Chandra band. Using the results for Gamma_X, we show that the alpha_{ox}-z relationship is unlikely to be a spurious result caused by redshifting of the observable X-ray spectral region. A correlation between alpha_{ox} and z implies evolution of the accretion process. We present a qualitative comparison of these new results with models for accretion disk emission.

Accepted by ApJ, 48 pages, 10 figures, 5 tables