  Certain quantum many-body ground states can be prepared by a large-depth
quantum circuit consisting of geometrically local gates. In the presence of
noise, local expectation values deviate from the correct value at most by an
amount comparable to the noise rate. This happens if the action of the
noiseless circuit, restricted to certain subsystems, rapidly mixes local
observables up to a small correction. The encoding circuit for the surface code
is given as an example.
