  The Heisenberg limit provides a quadratic improvement over the standard
quantum limit, and is the maximum quantum advantage that linear quantum sensors
could provide over classical methods. It remains elusive, however, because of
the inevitable presence of noise decohering quantum sensors. Namely, if
infinite rounds of quantum error correction corrects any part of a quantum
sensor's signal, a no-go result purports that the standard quantum limit
scaling can not be exceeded. We side-step this no-go result by using an optimal
finite number of rounds of quantum error correction, such that even if part of
the signal is corrected away, our quantum field sensing protocol can approach
the Heisenberg limit in the face of a linear rate of deletion errors. Here, we
focus on quantum error correction codes within the symmetric subspace.
Symmetric states of collective angular momentum are good candidates for
multi-qubit probe states in quantum sensors because they are easy to prepare
and can be controlled without requiring individual addressability. We show how
the representation theory of the symmetric group can inform practical decoding
algorithms for permutation-invariant codes. These decoding algorithms involve
measurements of total angular momentum, quantum Schur transforms or logical
state teleportations, and geometric pulse gates. Our completion of the theory
of permutation-invariant codes allows near-term implementation of quantum error
correction using simple quantum control techniques.
