Variational Spin-Squeezing Algorithms on Programmable Quantum Sensors

Arrays of atoms trapped in optical tweezers combine features of programmable analog quantum simulators with atomic quantum sensors. Here we propose variational quantum algorithms, tailored for tweezer arrays as programmable quantum sensors, capable of generating entangled states on demand for precision metrology. The scheme is designed to generate metrological enhancement by optimizing it in a feedback loop on the quantum device itself, thus preparing the best entangled states given the available quantum resources. We apply our ideas to the generation of spin-squeezed states on Sr atom tweezer arrays, where finite-range interactions are generated through Rydberg dressing. The complexity of experimental variational optimization of our quantum circuits is expected to scale favorably with system size. We numerically show our approach to be robust to noise, and surpassing known protocols.

Figure 1

Hybrid classical-quantum optimization on a programmable quantum sensor prepares spin-squeezed states for Ramsey interferometry. Variational control parameters $\mathit{\theta }$ generate trial states $|\mathrm{\Psi }(\mathit{\theta })⟩=\mathcal{S}(\mathit{\theta })R_y(\pi /2)|{{\downarrow }_z⟩}^{\otimes N}$. The squeezing parameter $\xi (\mathit{\theta })$ serves as cost function for classical optimization. Bottom: Optimization run on a $4\times 4$ array with $R_C/a=1.5$, using $\approx {10}^5$ simulated experimental runs, showing measured spin-squeezing of the trial states (blue dots, one iteration $=100$ measurements). A and B indicate theoretically obtainable squeezing with finite- and infinite-range one axis twisting [29, 46], respectively. Red line and orange error band represent the predicted squeezing minima at iteration $i$ of the search algorithm.

Figure 2

Quantum circuit representing the variational squeezing operation. The squeezer $\mathcal{S}(\mathit{\theta })$ is a sequence of $n$ unitary layers ${\mathcal{U}}_i$ ($i=1,\dots ,n$). Each ${\mathcal{U}}_i$ is further decomposed into two entangling interactions $D_z$ and $D_x$ and one global rotation $R_x$, as in Eq. (3), controlled by three variational parameters ${\tau }_i$, ${\vartheta }_i$, ${\tau }_i\prime$. Typical Husimi distributions [54] of the quantum state before each gate, are displayed on a generalized Bloch sphere. The action of the gates onto the permutation invariant subspace is indicated by red arrows.

Figure 3

Exact optimization results for circuit depths from $n=1$ to 5 layers, compared with FOAT [46] (dot dashed line). Top left panel: optimized squeezing parameter ${\xi }^2$ in a $4\times 4$ square array as a function of the interaction radius $R_c/a$. The vertical dashed line indicates the diagonal $R^{\star }=\sqrt{18}a$, while the horizontal ones show, from top to bottom, the squeezing obtainable with OAT, TAT and the fundamental squeezing limit ${\xi }_{\lim }^2(N)=[(2)/(N+2)]$. Top right panel: optimized squeezing parameter in a 1D array, as a function of the particle number $N$, at fixed $R_C/a=3$. Bottom left panel: optimization for a random realization, displayed in the inset, of a half-filled $6\times 6$ square array. The data points display averages over various random half-filling realizations, at the optimal pulse sequence. Bottom right panel: impact of normally distributed control noise fluctuating with a relative standard deviation ${\sigma }_{\text{noise}}$ around the optimal values.

Figure 4

Real-time dynamics of optimized pulse sequences, with different squeezer depths $n$ on a $4\times 4$ square array ($R_C/a=1.5$). Rotation pulses are treated as instantaneous, so that $t\in [0,{\sum }_{i=1}^n({\tau }_i+{\tau }_i^{\prime })]$ captures the cumulated interaction time. Dot-dashed lines show evolution under a single $D_z$ interaction. The displayed quantities are the (rotationally invariant) spin squeezing ${\xi }^2$ (top panel), quantum Fisher information $F_{\max }^Q/N$ (middle) and total angular momentum $⟨{\mathit{J}}^2⟩$ (bottom). The horizontal dashed lines in the top panel correspond to the spin squeezing obtainable with FOAT, infinite-range OAT and TAT, respectively, top to bottom. Vertical lines mark total interaction time for various circuit depths.