  Channel estimation is a critical task in multiple-input multiple-output
(MIMO) digital communications that substantially effects end-to-end system
performance. In this work, we introduce a novel approach for channel estimation
using deep score-based generative models. A model is trained to estimate the
gradient of the logarithm of a distribution and is used to iteratively refine
estimates given measurements of a signal. We introduce a framework for training
score-based generative models for wireless MIMO channels and performing channel
estimation based on posterior sampling at test time. We derive theoretical
robustness guarantees for channel estimation with posterior sampling in
single-input single-output scenarios, and experimentally verify performance in
the MIMO setting. Our results in simulated channels show competitive
in-distribution performance, and robust out-of-distribution performance, with
gains of up to $5$ dB in end-to-end coded communication performance compared to
supervised deep learning methods. Simulations on the number of pilots show that
high fidelity channel estimation with $25$% pilot density is possible for MIMO
channel sizes of up to $64 \times 256$. Complexity analysis reveals that model
size can efficiently trade performance for estimation latency, and that the
proposed approach is competitive with compressed sensing in terms of
floating-point operation (FLOP) count.
