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arXiv:2603.14095

Adaptive quantum metrology with large dynamic range using short one-axis twists

Tyler G. Thurtell, Akimasa Miyake

quant-phSubmitted Mar 14, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

Phase estimation with potentially large phase values, i.e., with large dynamic range, has many applications in quantum metrology, for example to atomic clocks. A recently proposed phase estimation scheme approaches the Heisenberg scaling in this global setting using sequences of increasingly squeezed Gaussian states as probes and adaptively chosen, potentially mid-circuit, measurements. In this work, we first observe that the pattern of increase in the squeezing of the probes is applicable even to states with some non-Gaussian features. We then propose an experimentally feasible version of this phase estimation scheme, based on the alternating application of one-axis twist (OAT) operations and rotations. Our protocols are explicitly described in terms of multiple OAT angles whose durations decrease polynomially with system size and spin-squeezing parameters that decay as $N^{-μ}$, with $μ>2/3$ in most cases. Using numerical computation of the system-size dependence $N^{-ν}$ of the Bayesian mean-squared error of an estimator, we show that these states are suitable for use in the phase estimation scheme, and highlight the protocols to achieve $ν=17/9$ and $53/27$ using two and three OAT operations respectively in the last adaptation stage. We also analyze the limited non-Gaussianity of the resulting probe states and discuss the role of non-Gaussianity in this protocol more generally. Finally, we analyze how robust these protocols are with respect to imperfections such as particle number fluctuations and coherent control fluctuations.

Comments
29 pages, 9 figures

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Phase estimation with potentially large phase values, i.e., with large dynamic range, has many applications in quantum metrology, for example to atomic clocks. A recently proposed phase estimation scheme approaches the Heisenberg scaling in this global setting using sequences of increasingly squeezed Gaussian states as probes and adaptively chosen, potentially mid-circuit, measurements. In this work, we first observe that the pattern of increase in the squeezing of the probes is applicable even to states with some non-Gaussian features. We then propose an experimentally feasible version of this phase estimation scheme, based on the alternating application of one-axis twist (OAT) operations and rotations. Our protocols are explicitly described in terms of multiple OAT angles whose durations decrease polynomially with system size and spin-squeezing parameters that decay as N−μN^{-μ}N−μ, with μ>2/3μ>2/3μ>2/3 in most cases. Using numerical computation of the system-size dependence N−νN^{-ν}N−ν of the Bayesian mean-squared error of an estimator, we show that these states are suitable for use in the phase estimation scheme, and highlight the protocols to achieve ν=17/9ν=17/9ν=17/9 and 53/2753/2753/27 using two and three OAT operations respectively in the last adaptation stage. We also analyze the limited non-Gaussianity of the resulting probe states and discuss the role of non-Gaussianity in this protocol more generally. Finally, we analyze how robust these protocols are with respect to imperfections such as particle number fluctuations and coherent control fluctuations.
Quantum metrology and sensing seek advantages in estimating an unknown parameter of some quantum state or channel, using entanglement such as spin squeezing produced by one-axis twists or other quantum resources. In particular, qubit phase estimation, or rotation sensing, appears as a ubiquitous problem with applications to electric field sensing, magnetometry, atomic clocks, and gyroscopes. By adopting the Bayesian formalism to the phase estimation problem to account for limited initial knowledge about the value of the phase, we formulate variational metrology and treat the state preparation (or encoding) and measurement (or decoding) procedures as parameterized quantum circuits. It is important to understand how effective various parametrized protocols are as well as how robust they are to the effects of complex noise such as spatially correlated noise. First, we propose a family of parametrized encoding and decoding protocols called arbitrary-axis twist ansatzes, and show that it can lead to a substantial reduction in the number of one-axis twists needed to achieve a target estimation error. Furthermore, we demonstrate that the estimation error associated with these strategies decreases with system size in a faster manner than classical (or no-twist) protocols, even in the less-explored regimes where the prior information is limited. Last, using a polynomial-sized tensor network algorithm, we numerically analyze practical variational metrology beyond the symmetric subspace of a collective spin and find that quantum advantage persists for the arbitrary-axis twist ansatzes with a few one-axis twists and smaller total twisting angles for practically relevant noise levels. Published by the American Physical Society 2024
Twist-untwist protocols for quantum metrology consist of a serial application of (1) unitary nonlinear dynamics (e.g., spin squeezing or Kerr nonlinearity), (2) parameterized dynamics $U(\ensuremath{\phi})$ (e.g., a collective rotation or phase space displacement), and (3) time reversed application of step 1. Such protocols are known to produce states that allow Heisenberg scaling for experimentally accessible estimators of $\ensuremath{\phi}$ even when the nonlinearities are applied for times much shorter than required to produce Schr\"odinger cat states. In this work, we prove that, asymptotically in the number of particles, twist-untwist protocols provide the lowest estimation error among quantum metrology protocols that utilize two calls to a weakly nonlinear evolution and a readout involving only first and second moments of a total spin operator $\stackrel{P\vec}{n}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{J}$. We consider the following physical settings: all-to-all interactions generated by one-axis twisting Jz2{J}_{z}^{2}Jz2​ (e.g., interacting Bose gases), constant finite range spin-spin interactions of distinguishable or bosonic atoms (e.g., trapped ions or Rydberg atoms, or lattice bosons). In these settings, we further show that the optimal twist-untwist protocols asymptotically achieve 85% and 92% of the respective quantum Cram\'er-Rao bounds. We show that the error of a twist-untwist protocol can be decreased by a factor of LLL without an increase in the noise of the spin measurement if the twist-untwist protocol can be noiselessly iterated as an LLL layer quantum alternating operator ansatz.
We propose two-stage optimized strategies for full-period optical phase estimation with single-mode Gaussian states and Gaussian measurements under a fixed energy constraint. In the first stage (Stage I), displaced squeezed probes and heterodyne measurements provide coarse localization of the phase to a window on the circle. In the second stage (Stage II), squeezed-vacuum probes with adaptive homodyne measurements perform efficient phase estimation inside the selected window. We derive a generalized Cramér-Rao bound for this family of two-stage Gaussian strategies, which contains the contribution from local parameter estimation in Stage II plus an overshoot penalty from coarse localization errors in Stage I. For E <= 25 photons and squeezing limited to 12 dB, protocols using displaced squeezed states in Stage I reduce the optimized two-stage bound relative to protocols using coherent states in Stage I, and remain within a factor of 3 to 30 of the idealized local squeezed-vacuum quantum Cramér-Rao bound.
Quantum metrology studies the use of quantum mechanical systems as measurement devices or sensors. Surprisingly, preparing a sensor in an entangled state can enhance measurement precision. The simplest protocols for entanglement enhancement sense only small changes in a quantity. The range of values over which a measurement protocol works is called its dynamic range. For many types of sensors we require end-to-end protocols that describe how to use entanglement to achieve enhanced precision over a large dynamic range. In this dissertation, we describe two approaches to achieving entanglement-enhanced sensing over a large dynamic range. The first approach uses entangling resources in a way that circumvents the dynamic-range limitation by numerically optimizing variational circuits. Our protocols show improved performance at fixed circuit depth compared to previous variational-circuit-based approaches, and the generality of our approach reveals several generic features of such protocols. The second approach uses adaptivity to first obtain a preliminary estimate of the quantity and then iteratively refines it. In particular, we describe the use of one-axis twisting dynamics at each stage of this protocol to minimize the expected estimation error.
We propose a quantum-enhanced metrology protocol known as partially concurrent twisting and sensing (PCTS) to effectively utilize time for high-precision sensing. In contrast to the traditional approach of separated twisting and sensing (STS), we concurrently execute twisting and sensing at the outset. Here, we explore one-axis twisting (OAT) over a short evolution period to induce spin squeezing, which has been used in several quantum-enhanced metrology. In particular for tiny estimated parameter, we analytically give the measurement precision of PCTS and demonstrate its metrological advantages compared with STS. Especially for modest particle number and OAT interaction strength, PCTS makes better use of entanglement generation time and achieves a larger metrological gain within the same total time. We estimate that our PCTS may offer up to 2 dB of metrological advantage over STS in realistic experiments with Bose condensed atoms or trapped ions. Although the measurement precision of PCTS may diminish with increasing estimated parameters, we can employ two hybrid classical-PCTS compensation techniques to facilitate high-precision measurements across a wide dynamic range. Furthermore, our approach eliminates the need to switch the estimated field, thereby reducing experimental demands and offering extensive application potential in atomic clocks and magnetometers.
Quantum-enhanced sensing with atomic ensembles has predominantly focused on qubit-based protocols, despite the growing ability of many experimental platforms to coherently control and entangle multi-level systems. Here, we investigate quantum-enhanced sensing with qutrit ensembles by introducing three experimentally feasible qutrit generalisations of the one-axis twisting (OAT) model that involve entangling operations only between two levels, while the third level primarily acts as a spectator. We characterize the metrological utility of the dynamically generated states using the quantum Fisher information toolbox. We find numerically that all three models offer considerable freedom in encoding direction for quantum-enhanced sensing, with up to 6 out of 8 possible directions exhibit near-Heisenberg scaling after a short evolution time. We discuss experimental access to this enhanced metrological precision via effective time-reversal protocols. Furthermore, we examine the practical issues of estimation ambiguity and local dissipation, and show that they can be largely overcome by optimizing the sensor operating point. Finally, we show that the measurement incompatibility in estimating multiple parameters simultaneously encoded in different directions with near-Heisenberg scaling of precision is suppressed at the zero operating point as the system size increases. In the process, we find that one of the models enables near-Heisenberg scaling metrology with a pair of commuting generators, a possibility that arises from the su(3)su(3)su(3) algebra and is thus absent in qubit-ensemble based sensors of collective SU(2)SU(2)SU(2) rotations.