Active B0 shimming involves setting currents through a set of shim coils to minimize spatial heterogeneity of B0 over a prescribed region. This is often phrased as a least-squares problem, based on an acquired field map. Noise in the field map "propagates" and appears as spatially dependent noise in the shimmed field. We show that the shim coils can be characterized by a noise propagation matrix, which is an intrinsic property of the coils and plays an analogous role to the g-factor in sensitivity encoding. This provides a useful metric for investigating the effect of noise in active shimming.