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arXiv:2610.30686

Time-reversal-based quantum metrology in a central-spin architecture: from SATIN to butterfly echo

Aditya Kumar, Sayan Choudhury

quant-phcond-mat.quant-gascond-mat.stat-mechSubmitted Oct 22, 2026• Announced Oct 23, 2026• 0 citations
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Abstract

We investigate quantum-enhanced time-reversal-based metrology using the central spin model, in which a single ancilla spin of magnitude $S_{\rm cen}$ mediates interactions between $N$ bath spins through a star-graph topology. We demonstrate that the SATIN (Signal Amplification Through Time-reversal Interaction) echo protocol can be implemented in this setup to achieve a Fisher information of $N^2/e$ with the collective $\langle J_y\rangle$ readout, corresponding to a $4.34\,\mathrm{dB}$ offset from the Heisenberg limit. Furthermore, we establish that this platform provides a pathway to perform the butterfly echo protocol by conjugating each evolution step with a random collective bath rotation, thereby achieving axis-agnostic Heisenberg-scaled sensing. Crucially, with a small $N-$independent number of random steps, the butterfly echo reaches a Fisher information of $N^2/4$ ($6\,\mathrm{dB}$ below the Heisenberg limit) for any unknown encoding axis, with the plateau gain tracking the Haar limit. Finally, we demonstrate that the central-spin magnitude $S_{\rm cen}$ provides a powerful knob to reduce the optimal protocol timescale without altering the metrological ceiling or the circuit depth required to reach isotropy. The proposed scheme is relevant to cavity QED and NMR quantum processors, where the central spin architecture arises naturally.

Comments
15 pages, 8 figures

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We investigate quantum-enhanced time-reversal-based metrology using the central spin model, in which a single ancilla spin of magnitude ScenS_{\rm cen}Scen​ mediates interactions between NNN bath spins through a star-graph topology. We demonstrate that the SATIN (Signal Amplification Through Time-reversal Interaction) echo protocol can be implemented in this setup to achieve a Fisher information of N2/eN^2/eN2/e with the collective ⟨Jy⟩\langle J_y\rangle⟨Jy​⟩ readout, corresponding to a 4.34 dB4.34\,\mathrm{dB}4.34dB offset from the Heisenberg limit. Furthermore, we establish that this platform provides a pathway to perform the butterfly echo protocol by conjugating each evolution step with a random collective bath rotation, thereby achieving axis-agnostic Heisenberg-scaled sensing. Crucially, with a small N−N-N−independent number of random steps, the butterfly echo reaches a Fisher information of N2/4N^2/4N2/4 (6 dB6\,\mathrm{dB}6dB below the Heisenberg limit) for any unknown encoding axis, with the plateau gain tracking the Haar limit. Finally, we demonstrate that the central-spin magnitude ScenS_{\rm cen}Scen​ provides a powerful knob to reduce the optimal protocol timescale without altering the metrological ceiling or the circuit depth required to reach isotropy. The proposed scheme is relevant to cavity QED and NMR quantum processors, where the central spin architecture arises naturally.
Quantum-enhanced metrology relies on entanglement to achieve sensitivities beyond the standard quantum limit. While remarkable progress has been made in generating highly entangled many-body states, extracting their metrological advantage remains a central challenge because the encoded information is often inaccessible to realistic measurements. A key development of the past decade has been the realization that many-body interactions can play a dual role: they can be used not only to generate entanglement, but also to decode it. This idea underlies interaction-based readout and time-reversal protocols, in which controlled non-linear dynamics transform weakly encoded signals into experimentally accessible observables. Cavity quantum electrodynamics (QED) provides a particularly powerful setting for these approaches because it combines collective enhancement, tunable interactions, and controllable reversibility within a single platform. In this review, we discuss the emergence of time-reversal protocols in cavity QED, from their conceptual roots in Loschmidt echoes to modern implementations of signal amplification through a time-reversed interaction (SATIN), scrambling-enhanced metrology, and more general interaction-based readout schemes. We examine the physical mechanisms that enable reversible many-body dynamics, review key experimental demonstrations, and discuss future directions involving complex entangled states, nonlinear decoding, and emerging quantum platforms. Together, these developments suggest that the ability to decode quantum information may become as important as the ability to generate it, establishing reversible many-body dynamics as a central resource for quantum-enhanced sensing.
Quantum sensing promises measurement precision beyond classical limits, but its practical realization is often hindered by decoherence and the challenges of generating and stabilizing entanglement in large-scale systems. Here, we experimentally demonstrate a scalable, scrambling-enhanced quantum sensing protocol, referred to as butterfly metrology, implemented on a cross-shaped superconducting quantum processor. By harnessing quantum information scrambling, the protocol converts local interactions into delocalized metrologically useful correlations, enabling robust signal amplification through interference of the scrambled and polarized quantum states. We validate the time-reversal ability via Loschmidt echo measurements and quantify the information scrambling through out-of-time-ordered correlators, establishing the essential quantum resources of our protocol. Our measurements reveal that the sensing sensitivity surpasses the standard quantum limit (SQL) with increasing qubit number, reaching 3.78 in a 9-qubit configuration, compared to the SQL of 3.0. The scheme further exhibits inherent robustness to coherent control errors and probed signal noise. This work demonstrates a readily scalable path toward practical quantum sensing advantages with prevalent experimental platforms.
The extreme sensitivity of chaotic systems to external perturbations makes them natural candidates for sensing applications. We propose a single-shot echo-based protocol for estimating small rotations about an unknown axis that leverages random symmetric probe states prepared via chaotic dynamics. In contrast to previous protocols for this axis-agnostic rotation sensing problem that depend on difficult-to-prepare anticoherent states, the random probe states used in our protocol can be prepared via constant-depth chaotic circuits composed of random one-axis twisting pulses. Further, the signal of interest can be extracted simply by measuring the total spin polarization. We demonstrate analytically that our protocol achieves Heisenberg scaling relative to an arbitrary rotation axis that need not be a priori known. We also investigate the effects of collective and single-particle dephasing in our protocol using analytical and numerical tools. While the requirements on dephasing rates to maintain Heisenberg sensitivity are strict, they are achievable in near-term experiments, for instance, in magnetometric rotosensing with high-spin lanthanide atoms such as dysprosium-164.
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.
We introduce a novel protocol, which enables Heisenberg-limited quantum-enhanced sensing using the dynamics of any interacting many-body Hamiltonian. Our approach - dubbed butterfly metrology - utilizes a single application of forward and reverse time evolution to produce a coherent superposition of a "scrambled" and "unscrambled" quantum state. In this way, we create metrologically-useful long-range entanglement from generic local quantum interactions. The sensitivity of butterfly metrology is given by a sum of local out-of-time-order correlators (OTOCs) - the prototypical diagnostic of quantum information scrambling. Our approach broadens the landscape of platforms capable of performing quantum-enhanced metrology; as an example, we provide detailed blueprints and numerical studies demonstrating a route to scalable quantum-enhanced sensing in ensembles of solid-state spin defects.
We propose a quantum metrology protocol based on a two-step joint evolution of the probe system and an ancillary qubit and quantum measurement. With a proper initial state of the ancillary qubit and an optimized evolution time, the quantum Fisher information (QFI) about the phase parameter encoded in the probe system is found to be determined by the expectation value of the square of a phase generator, irrespective of the probe initial state. Consequently, even if the probe is prepared as a finite-temperature state, faraway from the so-called resource state, e.g., the squeezed spin state or the Greenberger-Horne-Zeilinger state in atomic systems, the QFI in our protocol can approach the Heisenberg scaling N2{N}^{2}N2 with respect to the probe size NNN. This quadratic scaling behavior shows robustness against the imperfections about the initial state of the ancillary qubit and the optimized evolution time of the whole system. Also it is not sensitive to the deviation in system parameters and the qubit decoherence. Using the time-reversal strategy and a single-shot projective measurement, the classical Fisher information (CFI) in our metrology protocol is saturated with its quantum counterpart. Our work thus provides an economical method to reach the Heisenberg limit in metrology precision with no input of entanglement or squeezing.