Heisenberg-limited metrology with perturbing interactions

We show that it is possible to perform Heisenberg-limited metrology on GHZ-like states, in the presence of generic spatially local interactions during the measurement process. An explicit protocol, which relies on measurements and feedback based on polynomial-time classical computation, achieves the Heisenberg limit. In one dimension, matrix product state methods can be used to perform this classical calculation, while in higher dimensions the cluster expansion underlies the efficient calculations. The latter approach is based on an efficient classical sampling algorithm for short-time quantum dynamics, which may be of independent interest.