  Control using quantized feedback is a fundamental approach to system
synthesis with limited communication capacity. In this paper, we address the
stabilization problem for unknown linear systems with logarithmically quantized
feedback, via a direct data-driven control method. By leveraging a recently
developed matrix S-lemma, we prove a sufficient and necessary condition for the
existence of a common stabilizing controller for all possible dynamics
consistent with data, in the form of a linear matrix inequality. Moreover, we
formulate semi-definite programming to solve the coarsest quantization density.
By establishing its connections to unstable eigenvalues of the state matrix, we
further prove a necessary rank condition on the data for quantized feedback
stabilization. Finally, we validate our theoretical results by numerical
examples.
