Optimal metrology with variational quantum circuits on trapped ions
Christian D. Marciniak, Thomas Feldker, Ivan Pogorelov, Raphael Kaubruegger, Denis V. Vasilyev, Rick van Bijnen, Philipp Schindler, P. Zoller, R. Blatt, Thomas Monz
Cold atoms and ions exhibit unparalleled performance in frequency metrology epitomized in the atomic clock. More recently, such atomic systems have been used to implement programmable quantum computers and simulators with highest reported operational fidelities across platforms. Their strength in metrology and quantum information processing offers the opportunity to utilize tailored, programmable entanglement generation to approach the `optimal quantum sensor' compatible with quantum mechanics. Here we report quantum enhancement in metrology beyond squeezing through low-depth, variational quantum circuits searching for optimal input states and measurement operators in a trapped ion platform. We perform entanglement-enhanced Ramsey interferometry finding optimal parameters for variational quantum circuits using a Bayesian approach to stochastic phase estimation tailored to the sensor platform capabilities and finite dynamic range of the interferometer. We verify the performance by both directly using theory predictions of optimal parameters, and performing online quantum-classical feedback optimization to `self-calibrate' the variational parameters. In both cases we find that variational circuits outperform classical and direct spin squeezing strategies under realistic noise and imperfections. With 26 ions we achieve 2.02(8) dB of metrological gain over classical interferometers.
Cold atoms and ions exhibit unparalleled performance in frequency metrology epitomized in the atomic clock. More recently, such atomic systems have been used to implement programmable quantum computers and simulators with highest reported operational fidelities across platforms. Their strength in metrology and quantum information processing offers the opportunity to utilize tailored, programmable entanglement generation to approach the optimal quantum sensor' compatible with quantum mechanics. Here we report quantum enhancement in metrology beyond squeezing through low-depth, variational quantum circuits searching for optimal input states and measurement operators in a trapped ion platform. We perform entanglement-enhanced Ramsey interferometry finding optimal parameters for variational quantum circuits using a Bayesian approach to stochastic phase estimation tailored to the sensor platform capabilities and finite dynamic range of the interferometer. We verify the performance by both directly using theory predictions of optimal parameters, and performing online quantum-classical feedback optimization to self-calibrate' the variational parameters. In both cases we find that variational circuits outperform classical and direct spin squeezing strategies under realistic noise and imperfections. With 26 ions we achieve 2.02(8) dB of metrological gain over classical interferometers.
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.
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.
We study multiparameter sensing of two-dimensional (2D) and three-dimensional (3D) vector fields within the Bayesian framework for SU(2) quantum interferometry.We establish a method to determine the optimal quantum sensor, which establishes the fundamental limit on the precision of simultaneously estimating multiple parameters with an N -atom sensor.Keeping current experimental platforms in mind, we present sensors that have limited entanglement capabilities and yet significantly outperform sensors that operate without entanglement and approach the optimal quantum sensor in terms of performance.Furthermore, we show how these sensors can be implemented on current programmable quantum sensors with variational quantum circuits by minimizing a metrological cost function.The resulting circuits prepare tailored entangled states and perform measurements in an appropriate entangled basis to realize the best-possible quantum sensor given the native entangling resources available on a given sensor platform.Notable examples include a 2D and 3D quantum "compass" and a 2D sensor that provides a scalable improvement over unentangled sensors.Our results on optimal and variational multiparameter quantum metrology are useful for advancing precision measurements in fundamental science and ensuring the stability of quantum computers, which can be achieved through the incorporation of optimal quantum sensors in a quantum feedback loop.
Frequency metrology is a cornerstone of modern precision measurements and optical atomic clocks have emerged as one of the most precise measurement devices. In this progress report, we explore various Ramsey interrogation schemes tailored to optical atomic clocks primarily limited by laser noise. To incorporate frequency fluctuations directly into the theoretical model, we consider a Bayesian framework. In this context, we review fundamental bounds arising in Bayesian estimation theory, which serve as a benchmark throughout this work. We investigate the trade-off between entanglement-enhanced sensitivity and robustness against laser noise in order to identify optimal initial states, measurement schemes and estimation strategies. Beside standard protocols based on coherent spin states, spin-squeezed states and Greenberger-Horne-Zeilinger states, we consider variational Ramsey protocols implemented via low-depth quantum circuits based on one-axis twisting operations to approach optimal stability. In particular, we review known and identify new optimal Ramsey interrogation schemes for a variety of scenarios, including different experimental platforms, ensemble sizes and regimes characterized by a wide range of interrogation durations and dead times. Hence, this work establishes a comprehensive theoretical framework for optimizing Ramsey interrogation schemes, providing guidance for the development of next-generation optical atomic clocks.
Quantum entanglement offers powerful opportunities for enhancing measurement sensitivity beyond classical limits, with optical atomic clocks serving as a leading platform for such advances. This chapter introduces the principles of entanglement-enhanced quantum metrology and explores their applications to timekeeping. We review the theoretical framework of quantum phase estimation, comparing frequentist and Bayesian approaches, and discuss paradigmatic entangled states such as spin-squeezed and GHZ states. Particular emphasis is placed on the challenges posed by decoherence, which constrain the practical advantages that can be realized in large-scale devices. The discussion then turns to frequency estimation in atomic clocks, highlighting how experimental constraints shape the translation of abstract quantum limits into real performance gains. Finally, we outline emerging directions of contemporary quantum metrology. Together, these developments underscore the increasingly close interplay between quantum information processing and precision metrology.
We investigate optimized quantum state preparation for quantum metrology applications in noisy environments. Using the qfi-opt package, we simulate a low-depth variational quantum circuit (VQC) composed of a sequence of global rotations and entangling operations applied to a chain of qubits that are subject to dephasing noise. The parameters controlling the VQC are numerically optimized to maximize the quantum Fisher information, which characterizes the ultimte metrological sensitivity of a quantum state with respect to a global rotation. We find that, regardless of the details of the entangling operation implemented in the VQC, the optimal quantum states can be broadly classified into a trio of qualitative regimes, i.e., catlike, squeezedlike, and product states, associated with different dephasing rates. Our findings are relevant for designing optimal state-preparation strategies for next-generation quantum sensors exploiting entanglement, such as time and frequency standards and magnetometers, aimed at achieving state-of-the-art performance in the presence of noise and decoherence.