We present a quantization of a contact structure. This is based on a flat connection on a Hilbert bundle over the underlying contact manifold. The connection is built from a homological/BRST quantization of the dynamics of a strict ambient contact structure. Its two main ingredients is a tractor connection and a Heisenberg map. The former ingredient relies on treating the contact structure as a parabolic geometry. The latter is a symplectic analog of a Clifford structure. When this tractor connection admits a parallel section of the standard tractor bundle, our construction then gives a quantization of the corresponding Reeb dynamics. This result is a quantum mechanical analog of the relationship between parallel tractors, conformal geometries and Einstein metrics.