Time-reversal-based quantum metrology in a central-spin architecture: from SATIN to butterfly echo

We investigate quantum-enhanced time-reversal-based metrology using the central spin model, in which a single ancilla spin of magnitude $S_{\mathrm{c}\mathrm{e}\mathrm{n}}$ mediates interactions between $N$ bath spins through a star-graph topology. We demonstrate that the SATIN (Signal Amplification Through Time-reversal Interaction) echo protocol can be implemented in this setup to achieve a Fisher information of $N^2\mathrm{/}e$ with the collective $⟨J_y⟩$ readout, corresponding to a $4.34\mathrm{d}\mathrm{B}$ offset from the Heisenberg limit. Furthermore, we establish that this platform provides a pathway to perform the butterfly echo protocol by conjugating each evolution step with a random collective bath rotation, thereby achieving axis-agnostic Heisenberg-scaled sensing. Crucially, with a small $N-$independent number of random steps, the butterfly echo reaches a Fisher information of $N^2\mathrm{/}4$ ($6\mathrm{d}\mathrm{B}$ below the Heisenberg limit) for any unknown encoding axis, with the plateau gain tracking the Haar limit. Finally, we demonstrate that the central-spin magnitude $S_{\mathrm{c}\mathrm{e}\mathrm{n}}$ provides a powerful knob to reduce the optimal protocol timescale without altering the metrological ceiling or the circuit depth required to reach isotropy. The proposed scheme is relevant to cavity QED and NMR quantum processors, where the central spin architecture arises naturally.