Finite-round quantum error correction on symmetric quantum sensors

In quantum sensing using $N$ entangled probes, the variance of the estimated signal strength $\hat{\theta}$ scales like $ \Theta(N^{-2})$ at the Heisenberg limit, which is a quadratic improvement over the standard quantum limit, and is the maximum quantum advantage over classical methods. This limit remains elusive, however, because of the inevitable presence of noise decohering quantum sensors. Here, we introduce a quantum sensing protocol \texttt{ECSense} based on permutation-invariant quantum error correction (QEC) codes that support tunable code parameters to suit the physical noise model. We show that when the signal duration is much shorter than decoherence times, such that errors only accumulate during the $N$ qubit probe state preparation and idle stage, then the estimate's variance of $\Theta( N^{-3/2})$ is achievable in the presence of $\Theta(\sqrt{N})$ errors while the Heisenberg limit is achieved when the number of errors is a constant. In the more challenging setting where errors also occur during signal accumulation, we prove using a non-Markovian QEC strategy, that even for a linear number of deletion errors, a variance approaching the Heisenberg limit is still achievable. We illustrate a concrete way to implement our protocol on near-term quantum hardware using cavity-assisted geometric phase gates.