Quantum metrology via partial quantum error correction

We introduce a new method for error-corrected quantum metrology where only partial quantum error correction (QEC) is needed to suppress local noise and maintain the probe states' super-standard-quantum-limit (super-SQL) sensing performance. This stands in contrast to the existing QEC-assisted sensing schemes in Phys. Rev. Lett. 112, 080801 (2014) and Phys. Rev. Lett. 112, 150802 (2014), where a probe state is encoded into the logical subspace of a quantum code and error correction involves measurements on all checks of the code. Here, we encode the probe states into superpositions of energetically different states of the underlying quantum code. For our probe states, error correction using a subset of checks is enough to suppress noise both before and after phase imprinting. We analyze the tradeoff in noise suppression. For noise parallel to our phase imprinter of operator weight $l$, we achieve a suppression of $p^\delta$, where $p$ is the noise strength and $\delta = \lfloor (l+1)/2 \rfloor$. We propose an adaptive imprinter-weight-increasing strategy to maintain super-SQL performance as we scale up the system. In all our examples, checks and phase imprinters are chosen to be local operators, avoiding non-local connectivity.