Fig. 1: Schematic of practical quantum-enhanced sensing.

a The principle of quantum sensing with approximate quantum error correction (QEC). The errors \(\left\{{E}_{1},{E}_{2},...\right\}\) map the quantum states in the code space to disjoint subspaces and the recovery operations \(\left\{{R}_{1},{R}_{2},...\right\}\) covert the states back to the code space with the acquired phase being preserved. However, the recovered state in the code space is deformed due to the nature of the approximate QEC, as illustrated by the dashed arrows. b One example of practical quantum sensing protocol with a bosonic probe. The Wigner functions illustrate the evolution of the probe quantum state: When the probe state is initialized to \((\left|1\right\rangle +i\left|3\right\rangle )/\sqrt{2}\), the phase can be preserved even if there is a single-photon-loss error (a), which could be tracked and corrected via the recovery operation (R₁). c Schematic of the experimental setup for the quantum radiometry implemented with a superconducting architecture. The device is constructed with a bosonic probe that couples to a receiver mode.