  We provide the first analysis of a non-trivial quantization scheme for
compressed sensing measurements arising from structured measurements.
Specifically, our analysis studies compressed sensing matrices consisting of
rows selected at random, without replacement, from a circulant matrix generated
by a random subgaussian vector. We quantize the measurements using stable,
possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on
convex optimization. We show that the part of the reconstruction error due to
quantization decays polynomially in the number of measurements. This is in line
with analogous results on Sigma-Delta quantization associated with random
Gaussian or subgaussian matrices, and significantly better than results
associated with the widely assumed memoryless scalar quantization. Moreover, we
prove that our approach is stable and robust; i.e., the reconstruction error
degrades gracefully in the presence of non-quantization noise and when the
underlying signal is not strictly sparse. The analysis relies on results
concerning subgaussian chaos processes as well as a variation of McDiarmid's
inequality.
