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Physical Review Applied

Breaking the Measurement-Sensitivity–Maximum-Range Limit of Quantum Metrology Using Two Sequences

Yang Dong, Ce Feng, Shao-Chun Zhang, Y. H. Zheng, Xiang-Dong Chen, Guang‐Can Guo, Fang‐Wen Sun

Published Jun 7, 2023• 5 citations
Publisher

Abstract

In precision measurement and quantum metrology, the $2\ensuremath{\pi}$ periodicity inherent to any interferometric signal sets a fundamental limit in the simultaneous achievement of both high measurement sensitivity and range. Here, based on the extended period of the Lissajous curve, we develop a method to break this sensitivity--maximum-range limit completely. In the measurement of magnetic field using an electronic spin in diamond under ambient conditions, we extend the maximum range by 2 orders of magnitude without sacrificing detection sensitivity and bandwidth for sufficiently long measurement time. Furthermore, we achieve precise detection of a time-varying periodic waveform, where a large measurement range can induce high accuracy and demonstrate a Heisenberg-like scaling within the coherent regime. This Lissajous-curve-based range extension method can be directly applied to any other modulo-limited quantum interferometers, which makes a useful and indispensable stepping stone toward practical interferometric quantum metrology.

DOI
10.1103/physrevapplied.19.064022

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In precision measurement and quantum metrology, the 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ periodicity inherent to any interferometric signal sets a fundamental limit in the simultaneous achievement of both high measurement sensitivity and range. Here, based on the extended period of the Lissajous curve, we develop a method to break this sensitivity--maximum-range limit completely. In the measurement of magnetic field using an electronic spin in diamond under ambient conditions, we extend the maximum range by 2 orders of magnitude without sacrificing detection sensitivity and bandwidth for sufficiently long measurement time. Furthermore, we achieve precise detection of a time-varying periodic waveform, where a large measurement range can induce high accuracy and demonstrate a Heisenberg-like scaling within the coherent regime. This Lissajous-curve-based range extension method can be directly applied to any other modulo-limited quantum interferometers, which makes a useful and indispensable stepping stone toward practical interferometric quantum metrology.
periodicity of phase readout introduces a wrap-around ambiguity, which confines the unambiguous measurement range to a single period and leads to a structural trade-off between precision and measurement range in high-sensitivity settings. To address this problem, we propose a robust Chinese remainder theorem-based multi-baseline measurement scheme: we choose a set of pairwise coprime scale factors and recover the wrap-around number from congruence relations among the reduced phases. In the setting of optical-interferometric quantum telescopes, we construct a telescope array based on the Chinese remainder theorem using fractional baselines, which substantially enlarges the unambiguous angular range (effective field of view) while retaining the high precision enabled by long baselines. Numerical simulations show that, under the same constraint on the shortest achievable baseline, our scheme achieves an exponential improvement in detection range over existing methods and attains a better range-precision trade-off with fewer probing resources. This framework supports practical multi-baseline quantum-interferometric observations and can be generalized to other phase-sensitive quantum precision-measurement tasks.
Leveraging quantum effects in metrology such as entanglement and coherence allows one to measure parameters with enhanced sensitivity. However, time-dependent noise can disrupt such Heisenberg-limited amplification. We propose a quantum-metrology method based on the quantum-signal-processing framework to overcome these realistic noise-induced limitations in practical quantum metrology. Our algorithm separates the gate parameter φφφ~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter θθθ~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of 10−410^{-4}10−4 radians in standard deviation for learning θθθ in superconducting-qubit experiments, outperforming existing alternative schemes by two orders of magnitude. We also demonstrate the increased robustness in learning time-dependent gate parameters through fast Fourier transformation and sequential phase difference. We show both theoretically and numerically that there is an interesting transition of the optimal metrology variance scaling as a function of circuit depth ddd from the pre-asymptotic regime d≪1/θd \ll 1/θd≪1/θ to Heisenberg limit d→∞d \to \inftyd→∞. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter φφφ scales faster than the asymptotic Heisenberg limit as a function of depth, Var(φ^)≈1/d4\text{Var}(\hatφ)\approx 1/d^4Var(φ^​)≈1/d4. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.
Quantum metrology achieves Heisenberg-limited precision in single-parameter estimation, but its multiparameter extension is fundamentally constrained by both parameter-encoding and measurement incompatibility. Noncommuting signal generators may cause incompatible parameter-encoding, preventing the quantum Fisher information matrix from simultaneously achieving the Heisenberg scale for all parameters. Due to incompatible optimal measurements, the classical Fisher information matrix represents the practical attainable precision. Here, we introduce a multiparameter metrology framework based on indefinite evolution (IE), in which different control operations and signal reversal are placed in a coherent superposition. For a single-qubit probe with mutually orthogonal signal generators, IE enables compatible parameter encoding and optimal measurement without the signal reversal. For parallel generators, where only signal reversal realized by its generator is available, IE can achieve the same performance. We further extend this mechanism to noisy, many-body, and high-dimensional probes, and establish general conditions for achieving the simultaneous Heisenberg-limit. In contrast, definite evolution cannot achieve the same performance under compatible optimal measurements, even when signal reversal is available. Our results identify IE as an operational resource for overcoming multiparameter incompatibility and open a route toward attainable Heisenberg-limited sensing in interferometric platforms.
The ultimate precision of phase estimation is limited by the Heisenberg scaling Δϕ0=K/NΔϕ_0 = K/NΔϕ0​=K/N, where K∼1K\sim1K∼1 is a numerical prefactor and NNN is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity Δϕ0Δϕ_0Δϕ0​ and the width δϕδϕδϕ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that δϕδϕδϕ strongly depends on KKK. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range δϕ=π/2δϕ=π/2δϕ=π/2.
A key challenge of magnetometry lies in the simultaneous optimization of magnetic field sensitivity and maximum field range. In interferometry-based magnetometry, a quantum two-level system acquires a dynamic phase in response to an applied magnetic field. However, due to the 2π periodicity of the phase, increasing the coherent interrogation time to improve sensitivity reduces field range. Here we introduce a route towards both large magnetic field range and high sensitivity via measurements of the geometric phase acquired by a quantum two-level system. We experimentally demonstrate geometric-phase magnetometry using the electronic spin associated with the nitrogen vacancy (NV) color center in diamond. Our approach enables unwrapping of the 2π phase ambiguity, enhancing field range by 400 times. We also find additional sensitivity improvement in the nonadiabatic regime, and study how geometric-phase decoherence depends on adiabaticity. Our results show that the geometric phase can be a versatile tool for quantum sensing applications.
Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.