Zooming type adaptive quantizers have been introduced in the networked control literature as efficient coders for stabilizing open-loop unstable noise-free systems connected over noiseless channels with arbitrary initial conditions. Such quantizers can be regarded as a special class of the Goodman-Gersho adaptive quantizers. In this paper, we provide a stochastic stability result for such quantizers when the system is driven by an additive noise process. Conditions leading to stability are evaluated when the system is driven by noise with non-compact support for its probability measure. It is shown that zooming quantizers are efficient and almost achieve the fundamental lower bound of the logarithm of the absolute value of an unstable eigenvalue. In particular, such quantizers are asymptotically optimal when the unstable pole of the linear system is large for a weak form of stability.