Quantum Variational Optimization of Ramsey Interferometry and Atomic Clocks

We discuss quantum variational optimization of Ramsey interferometry with ensembles of $N$ entangled atoms, and its application to atomic clocks based on a Bayesian approach to phase estimation. We identify best input states and generalized measurements within a variational approximation for the corresponding entangling and decoding quantum circuits. These circuits are built from basic quantum operations available for the particular sensor platform, such as one-axis twisting, or finite range interactions. Optimization is defined relative to a cost function, which in the present study is the Bayesian mean squared error of the estimated phase for a given prior distribution; i.e., we optimize for a finite dynamic range of the interferometer. In analogous variational optimizations of optical atomic clocks, we use the Allan deviation for a given Ramsey interrogation time as the relevant cost function for the long-term instability. Remarkably, even low-depth quantum circuits yield excellent results that closely approach the fundamental quantum limits for optimal Ramsey interferometry and atomic clocks. The quantum metrological schemes identified here are readily applicable to atomic clocks based on optical lattices, tweezer arrays, or trapped ions. While in the present work variationally optimized circuits are found with classical simulations, optimization can also be performed “on” the (physical) quantum sensor, also in regimes not accessible to classical computations and in the presence of imperfections.

Popular Summary

Progress in physics closely correlates with the ability to make precise measurements. Conversely, newly gained insights and methods in physics can be used to develop improved and novel measurement instruments. Here, we take the latter approach and combine quantum-information concepts with quantum metrology to devise optimized—and indeed optimal—atomic sensors, such as atomic clocks and matter-wave interferometers.

Quantum entanglement provides an opportunity to reduce quantum fluctuations inherent in quantum measurements beyond what is possible with uncorrelated particles. However, quantum physics imposes ultimate limits on quantum sensing, thus defining “optimal” quantum sensors. We show that variational quantum circuits, which are built from surprisingly few quantum operations naturally available on a specific sensor platform, can be programmed to generate entangled input states and measurements close to optimality. Our results point to a viable experimental route for optimal quantum sensing on intermediate-scale quantum devices acting as “programmable quantum sensors.”

We envision that programming of variational quantum circuits can be performed not only within theoretical modeling but also “on device” as a quantum-classical feedback loop running on the physical quantum sensor itself. This will enable optimization in the presence of imperfections and decoherence as well as in the regime of many particles, where the complexity of the underlying quantum many-body problem exceeds the capabilities of classical computations.

Figure 1

(a) Quantum circuit representation of Ramsey interferometer with uncorrelated atoms. The phase $\varphi$ is imprinted on the atomic spin superposition prepared by global $\pi /2$ rotation around $y$ axis, ${\mathcal{R}}_y(\pi /2)$. Consequent rotation, ${\mathcal{R}}_x(\pi /2)$, and measurement of difference $m$ of atoms in eigenstates $|\uparrow ⟩$ and $|\downarrow ⟩$ in $z$ basis allows estimating the phase $\varphi$ using an estimator function ${\varphi }_{\mathrm{est}}(m)$. (b) Quantum circuit of a generalized Ramsey interferometer with generic entangling and decoding operations ${\mathcal{U}}_{\mathrm{en}}$ and ${\mathcal{U}}_{\mathrm{de}}$, respectively. Our variational approach (c) consists of an ansatz, where optimal ${\mathcal{U}}_{\mathrm{en}}$ and ${\mathcal{U}}_{\mathrm{de}}$ are approximated by low-depth circuits. These are built from “layers” of elementary operations, which are provided by the given platform. We specify the variationally optimized quantum sensor by circuits ${\mathcal{U}}_{\mathrm{en}}(\mathit{\theta })$ and ${\mathcal{U}}_{\mathrm{de}}(\mathbf{\vartheta })$ [see Eqs. (6) and (7)], of depth $n_{\mathrm{en}}$ and $n_{\mathrm{de}}$, respectively. Here $\mathit{\theta }\equiv \{{\theta }_i\}$ and $\mathbf{\vartheta }\equiv \{{\vartheta }_i\}$ are vectors of variational parameters to be optimized for a given strategy represented by a cost function $\mathcal{C}$ defined here as Bayesian mean squared error (BMSE) [see Eqs. (2) and (10)]. We illustrate the approach with a variational circuit built from global spin rotations ${\mathcal{R}}_x$ and one-axis-twisting gates ${\mathcal{T}}_{x,z}$ available in neutral atom and ion quantum simulation platforms, as discussed in Sec. 2b. The circuit’s optimization, shown as a feedback loop (in red), can be performed on a classical computer, or, if the complexity of underlying quantum many-body problem exceeds capabilities of classical computers, on the sensor itself, thus leading to a (relevant) quantum advantage; see Sec. 2g.

Figure 2

Performance of the variationally enhanced interferometer with $N=64$ particles. Performance is shown in terms of the posterior phase distribution width relative to the prior width, $\mathrm{\Delta }\varphi /\delta \varphi$, for a given prior, that is, for a given dynamic range of the interferometer. Colored lines show the performance of variationally optimized circuits for the depth $(n_{\mathrm{en}},n_{\mathrm{de}})$ of entangling and decoding layers as indicated. The number of variational parameters is given by $3(n_{\mathrm{en}}+n_{\mathrm{de}})$. The performance of the optimal quantum interferometer (OQI) [31] is indicated by the dotted line. The shaded areas indicate the classically accessible (purple) and the quantum mechanically forbidden (gray) regions (for $N=64$). Related results applied to atomic clocks are shown in Fig. 10.

Figure 3

Visualization of quantum states $|{\psi }_{\varphi }⟩=\exp (-i\varphi J_z)|{\psi }_{\mathrm{in}}⟩$, and quantum measurement operators as Wigner distributions on the generalized Bloch sphere for $N=64$ and $\delta \varphi \approx 0.7$. The first [(a),(d)], second [(b),(e)], and third [(c),(f)] column correspond to $(n_{\mathrm{en}},n_{\mathrm{de}})=(1,0)$ (squeezed input state, and $J_y$ measurement operator), the optimal quantum interferometer, and to a (1,3) quantum circuit, respectively. Measurement operators are visualized as colored contours on the Bloch sphere corresponding to different measurement outcomes. The corresponding optimized (optimal) states $|{\psi }_{\varphi }⟩$ are shown at various angles $\varphi$ as gray shaded areas. (a)–(c) Three-dimensional view of the generalized Bloch sphere with a state rotated to $\varphi =\pi /3$. (d)–(f) Top view of the Bloch sphere with the state rotated to angles $\varphi =0,\pi /3,2\pi /3$. (g) Measurement probability $p(m|\varphi )$ [see Eq. (9)] corresponding to the overlap between the contours of the measurement distribution and the respective state distribution, displayed in the same column. The three rows correspond to the above three angles $\varphi$. Note that for the $J_y$ measurement the distributions at angles $\pi /3$ and $2\pi /3$ are indistinguishable in measurement statistics. In contrast, for the OQI and the (1,3) quantum circuit these angles are well resolved.

Figure 4

(a) Phase $\varphi$ dependence of the estimator expectation value [Eq. (11)] of the optimized $N=64$ particle interferometer at different circuit depths $(n_{\mathrm{en}},n_{\mathrm{de}})$ with $3(n_{\mathrm{en}}+n_{\mathrm{de}})$ variational parameters, in comparison to the optimal quantum interferometer (OQI). The optimization is performed for the prior distribution width $\delta \varphi \approx 0.7$, indicated by the vertical lines. (b) Mean squared error [Eq. (8)] corresponding to the estimator expectation values curves above.

Figure 5

Relative performance of the covariant, phase operator based interferometer (POI) (blue) and the variational (1,3) and (2,5) interferometers (green and red points, respectively) with respect to the OQI for a given system size $N$. The $\chi$ ratio is defined in Eq. (12), OQI corresponds to $\chi =1$. The dashed lines represent empirical scalings, green and red one scale as $\sim N$, and blue is $\sim N^{-0.77}$.

Figure 6

Accuracy of an optimized $N=64$ particle interferometer in the presence of single-particle dephasing noise exposures $\gamma T/\delta \varphi$ relative to the prior distribution width, indicated by different line styles. The results are displayed for different circuit complexities $(n_{\mathrm{en}},n_{\mathrm{de}})$ with $3(n_{\mathrm{en}}+n_{\mathrm{de}})$ variational parameters. For comparison the accuracy of the noise free optimal quantum interferometer (OQI) is indicated by the black dotted line.

Figure 7

Plot of $\mathrm{\Delta }{\varphi }_{M}N$, i.e., the standard deviation of an effective measurement rescaled by the ensemble size $N$, versus prior width $\delta \varphi$. Solid lines show results for the optimized interferometer with circuit depth $(n_{\mathrm{en}},n_{\mathrm{de}})=(2,5)$ in comparison to the analytic expression describing a GHZ state interferometer [Eq. (21)] shown with dashed lines and the $\pi$-corrected Heisenberg limit including phase slips [Eq. (22)] shown with dotted lines. The Heisenberg limit and the $\pi$-corrected Heisenberg limit are indicated with dotted horizontal lines.

Figure 8

Performance of the variationally optimized $N=16$ Ramsey interferometer on a $4\times 4$ lattice interacting via finite range Rydberg dressing interactions (23) with interaction radii $R_C$ in units of the array spacing $a$. Colored lines show the reduction of the posterior phase distribution width $\mathrm{\Delta }\varphi$ relative to the prior distribution width $\delta \varphi$ for variationally optimized circuits complexity $(n_{\mathrm{en}},n_{\mathrm{de}})$ with $3(n_{\mathrm{en}}+n_{\mathrm{de}})$ variational parameters. The performance of the optimal quantum interferometer (OQI)[31] is indicated by dotted lines. The shaded areas indicate the classically accessible (purple) and the quantum mechanically inaccessible (gray) regions. (a) Prior width dependence of the optimized solution at $R_C/a=2$. (b) Interaction range dependence of the optimal solution at a prior distribution width $\delta \varphi \approx 0.8$ indicated by the vertical dashed line in (a).

Figure 9

Allan deviation representing a single run of a flicker-frequency-noise-limited clock servo loop based on variationally optimized $N=64$ particle interferometers at different circuit complexities $(n_{\mathrm{en}},n_{\mathrm{de}})$ and a Ramsey interrogation time $b_{2}T\approx 0.5$ (solid lines). For comparison the long time scaling predicted by Eq. (25) is shown by the dashed lines.

Figure 10

Dimensionless Allan deviation $\sigma$ [see Eq. (25)] of a $N=64$ flicker-frequency-noise-limited clock at constant averaging time $\tau$ as a function of the Ramsey interrogation time $T$ rescaled to the bandwidth of the laser noise. The dotted black line indicates the instability of the optimal quantum clock (OQC). (a) Analytic expressions for the instability of coherent spin state (CSS) and GHZ state clock (solid lines) in comparison to quantum projection noise limits, namely the standard quantum noise limit (SQL), $\pi$-corrected Heisenberg limit ($\pi \mathrm{HL}$), HL (dashed lines), and the coherence time limits (CTL) of a CSS clock and the OQC (densely dotted lines). (b) Variational approximation of the OQC at increasing circuit complexities $(n_{\mathrm{en}},n_{\mathrm{de}})$ with $3(n_{\mathrm{en}}+n_{\mathrm{de}})$ variational parameters. Markers show the numerically determined instability extracted from simulations of the full feedback loop of an atomic clock, described in detail in Appendix pp7, for a selection of optimized protocols. The numerical data are displayed only up to values of $b_{2}T$ where no fringe hop occurred within the $2\times {10}^6$ simulated clock cycles. Beyond this point an abrupt loss of stability was observed.

Figure 11

Particle number $N$ dependence of the dimensionless Allan deviation of a flicker-frequency-noise-limited clock at a constant averaging time $\tau$ and the optimal Ramsey interrogation time for different circuit complexities $(n_{\mathrm{en}},n_{\mathrm{de}})$ with $3(n_{\mathrm{en}}+n_{\mathrm{de}})$ variational parameters. For comparison we also show the performance of the optimal quantum clock (dotted line) and the asymptotic behavior [see Eq. (28)] obtained from the $\pi$-corrected HL (dashed line).

Figure 12

Largest fraction $T_{D,\max }/T_C$ of dead time compared to cycle time for which the clock operation is limited predominantly by the variationally optimized instability displayed in Fig. 11. The overall instabilities at dead time fractions above the lines are limited by the Dick effect instead.

Figure 13

(a) Recalculation of the particle number dependence of the dimensionless Allan deviation in Fig. 11 for constrained interaction angles. (b) The cumulative angle of all one-axis-twisting gates ${\mathcal{T}}_{x,y}$ required to obtain the dimensionless Allan deviations displayed above. The vertical dashed line indicates the interaction angle of $\pi /2$ required to prepare a GHZ state.