Finite-round quantum error correction on symmetric quantum sensors

The Heisenberg limit provides a quadratic improvement over the standard quantum limit, and is the maximum quantum advantage that quantum sensors could provide over classical methods. This limit 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 using Markovian quantum error correction. We side-step this no-go result and prove that in the limit of a large number of qubits, our quantum field sensing protocol has a precision that approaches the Heisenberg limit despite a linear rate of deletion errors. This is achieved by using an optimally determined, finite number of rounds of quantum error correction married with adaptive, non-Markovian signal recovery procedures. Our protocol is based on permutation invariant quantum error correction codes, which can be designed to admit correction for a tunable number of bit-flip, phase-flip, and deletion errors. We discuss near-term implementations using quantum control assisted by coupling the spins to a common bosonic mode.