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 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 action angles $O(1/N)$ 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.

Figure 1

Measurement precision $\mathrm{\Delta }\eta$ as a function of number of spins (log-log scale). Shown are the shot-noise limit (upper red dashed line), the quantum Cramér-Rao bound given by Eq. (2) (lower red dashed line), and this protocol (blue). Inset: Fidelity for sets of ensemble sizes using the same number of Grover steps, which grow as $N^{1/4}$.

Figure 2

Suppression of dephasing via dynamical decoupling inherent in the sequence of GPGs used for each of the operators $U_s$ and $U_w$. Solid curves are filter functions using the GPGs. Dashed curves are plots of Eq. (6), which is a good approximation for $\omega /g<1/\pi N$. Short dashed curves are the bare case without decoupling. Red, green, and blue curves correspond to $n=(10,100,1000)$ spins.

Figure 3

Precision obtained for 70 spins with a single measurement of ${J^z}^2$ as a function of mode decay for various global dephasing factors $A(T)$: no global dephasing (blue line), $A(T)={10}^{-6}$ (blue dashed line), $A(T)={10}^{-5}$ (blue dot-dashed line), $A(T)={10}^{-4}$ (blue dotted line). These dephasings correspond to an underlying decoherence rate of ${\gamma }_{\mathrm{GDP}}={10}^{-4}g$ accumulated over each phasing gate of duration $T$. For a zero temperature Ohmic bath, the corresponding cuttoff frequencies are ${\omega }_c/g=\{0.003,0.022,0.094\}$. Black line shows performance without dynamical decoupling with $A_0(T)=0.0223$.