  In this paper we study the quantization stage that is implicit in any
compressed sensing signal acquisition paradigm. We propose using Sigma-Delta
quantization and a subsequent reconstruction scheme based on convex
optimization. We prove that the reconstruction error due to quantization decays
polynomially in the number of measurements. Our results apply to arbitrary
signals, including compressible ones, and account for measurement noise.
Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli)
random compressed sensing measurements, as well as for both high bit-depth and
coarse quantizers, and they extend to 1-bit quantization. In the noise-free
case, when the signal is strictly sparse we prove that by optimizing the order
of the quantization scheme one can obtain root-exponential decay in the
reconstruction error due to quantization.
