Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.

Mar 27, 20251 citation

Quantum metrology with time-dependent Hamiltonians

Andrew N. Jordan, Shengshi Pang, Jing Yang

Bulletin of the American Physical SocietyMar 7, 20180 citations

Quantum Metrology: How Close Can We Get to the Ideal Heisenberg Limit Under Realistic Noise?

Deepinder Sidhu

Quantum metrology promises Heisenberg-limited precision, yet every practical platform exhibits a breakdown of this scaling as resources increase. We present a compact, platform-independent framework in which the total sensing resource is decomposed as N = M L, where M denotes the number of probes interrogated in parallel and L denotes the coherent sequential interrogation depth. Under realistic decoherence, sequential phase accumulation incurs an exponential noise penalty that imposes a finite bound on usable coherent depth, thereby terminating ideal Heisenberg scaling. We show that the resulting Heisenberg-saturation knee is not a plotting artifact or a channelspecific bound, but the direct consequence of constrained optimization under a fixed coherence budget. For fixed total resource N, contour plots in the (M, L) plane reveal a saddle-point structure at which achievable precision is minimized and beyond which additional resources cannot be converted into coherent phase sensitivity. This geometric interpretation yields a concrete design rule: quantum advantage is maximized not by increasing N indiscriminately, but by optimally allocating resources between parallelization and coherent depth.Quantum metrology promises Heisenberg-limited precision, yet every practical platform exhibits a breakdown of this scaling as resources increase. We present a compact, platform-independent framework in which the total sensing resource is decomposed as N = M L, where M denotes the number of probes interrogated in parallel and L denotes the coherent sequential interrogation depth. Under realistic decoherence, sequential phase accumulation incurs an exponential noise penalty that imposes a finite bound on usable coherent depth, thereby terminating ideal Heisenberg scaling. We show that the resulting Heisenberg-saturation knee is not a plotting artifact or a channelspecific bound, but the direct consequence of constrained optimization under a fixed coherence budget. For fixed total resource N, contour plots in the (M, L) plane reveal a saddle-point structure at which achievable precision is minimized and beyond which additional resources cannot be converted into coherent phase sensitivity. This geometric interpretation yields a concrete design rule: quantum advantage is maximized not by increasing N indiscriminately, but by optimally allocating resources between parallelization and coherent depth.

Jul 18, 20260 citations

Quantum Metrology in the Presence of Quantum Oscillations

Seyed Mohammad Hosseiny, Jamileh Seyed‐Yazdi, Milad Norouzi +1 more

International Journal of Theoretical PhysicsMar 19, 20243 citations

Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing

Yulong Dong, Jonathan A. Gross, Murphy Yuezhen Niu

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 $φ$ ~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter $θ$ ~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of $10^{-4}$ radians in standard deviation for learning $θ$ 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 $d$ from the pre-asymptotic regime $d \ll 1/θ$ to Heisenberg limit $d \to \infty$ . Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter $φ$ scales faster than the asymptotic Heisenberg limit as a function of depth, $\text{Var}(\hatφ)\approx 1/d^4$ . Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.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 $φ$~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter $θ$~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of $10^{-4}$ radians in standard deviation for learning $θ$ 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 $d$ from the pre-asymptotic regime $d \ll 1/θ$ to Heisenberg limit $d \to \infty$. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter $φ$ scales faster than the asymptotic Heisenberg limit as a function of depth, $\text{Var}(\hatφ)\approx 1/d^4$. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.

arXiv:2209.11207Sep 22, 20221 citation

φ

~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter

θ

~(swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of

10^{-4}

radians in standard deviation for learning

θ

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

d

from the pre-asymptotic regime

d \ll 1/θ

to Heisenberg limit

d \to \infty

. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter

φ

scales faster than the asymptotic Heisenberg limit as a function of depth,

\text{Var}(\hatφ)\approx 1/d^4

. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.