A geometric pathway to scalable quantum sensing

Entangled resources enable quantum sensing that achieves Heisenberg scaling, a quadratic improvement on the standard quantum limit, but preparing large N spin entangled states is challenging in the presence of decoherence. We present a quantum control strategy using highly nonlinear geometric phase gates which can be used for generic state or unitary synthesis on the Dicke subspace with O(N) or O(N^2) gates, respectively. The method uses a dispersive coupling of the spins to a common bosonic mode and does not require addressability, special geometric layouts or detunings, or interactions between the spins. By using amplitude amplification our control sequence for preparing states ideal for metrology can be significantly simplified to O(N^5/4) geometric phase gates with size O(1/N) action angles that are more robust to mode decay. The geometrically closed path of the control operations ensures the gates are insensitive to the initial state of the mode and the sequence has built-in dynamical decoupling providing resilience to dephasing errors. We describe implementations using trapped ions or Rydberg atoms and show how to prepare quantum error correction code words in the Dicke space.