Recovering optimal precision in quantum sensing with time domain imperfections

TL;DR

This work tackles frequency estimation in quantum sensing under clock-domain imperfections that effectively average dynamics in time, focusing on a non-Markovian environment. It introduces a control-enhanced (CE) metrology protocol that interposes a small set of intermediate pulses and an ancilla to counteract clock-uncertainty-induced bias, showing a recovery of Heisenberg-like scaling up to a hardware-limited term. Through biased Cramér-Rao analysis and explicit quantum Fisher information bounds, the authors contrast FE and CE strategies, deriving tight bounds and revealing how CE can approach interaction-free performance in the finite-repetition and large-data limits. The theoretical results are corroborated by nuclear magnetic resonance experiments demonstrating that, even with imperfect control, the CE protocol outperforms control-free methods across SWAP and CNOT interaction models, highlighting the practical robustness of quantum-control-assisted sensing in realistic imperfection regimes. The work additionally connects ergodicity considerations and autonomous clocks to the observed metrological gains, suggesting broader implications for quantum sensing under time-domain imperfections.

Abstract

Quantum control plays a crucial role in enhancing precision scaling for quantum sensing. However, most existing protocols require perfect control, even though real-world devices inevitably have control imperfections. Here, we consider a fundamental setting of quantum sensing with time domain imperfections, where the duration of control pulses and the interrogation time are all subject to uncertainty. Under this scenario, we investigate the task of frequency estimation in the presence of a non-Markovian environment. We design a control strategy and prove that it outperforms any control-free strategies, recovering the optimal Heisenberg limit up to a small error term that is intrinsic to this model. We further demonstrate the advantage of our control strategy via experiments on a nuclear magnetic resonance (NMR) platform. Our finding confirms that the advantage of quantum control in quantum sensing persists even in the presence of imperfections.

Figures (7)

• Figure 1: The clock uncertainty model and the control strategy. We consider a generic model of clock uncertainty where any time interval, set by the experimenter using an imperfect stopwatch, is subject to some random deviation [see (a)]. Consequently, the total interrogation time [see (b)], the duration of control pulses and the length of the interval between any consecutive pulses [(c)] are all subject to uncertainty. Under this model, we propose a control strategy to enhance the performance of frequency estimation [see (d)], where the system (S) is coupled to an environment (E) and an ancillary qubit (A). The dashed arrows between (c) and d) indicate the timing of inserting two intermediate control pulses (grey boxes in the middle).
• Figure 2: Numerical results for finite $\nu$ and the CE strategy.(a) Numerical results for both the FE case and the CE case with a uniform $f$: The IF case corresponds to setting the system-environment interaction strength to zero. The heights of the shaded areas indicate the standard deviation resulting from different choices of $\omega$. Note that in some instances, the error in the CE case is even smaller than in the IF case, which can be attributed to the CE case exhibiting a larger bias compared to the IF scenario that accidentally benefits the estimation in the non-asymptotic regime. (b) The circuit representation of our CE strategy: The gates $H,~S,$ and$~X$ are the Hadamard gate, the $\pi/2$-phase gate, and the Pauli-$X$ gate, respectively. The unitary $U_{\mathsf{FE}}$ is the free evolution under the total Hamiltonian. The lengths of the intervals between the control operations are $t_1=\pi/(4g)$ and $t_2=\pi/(2g)$, the initial state is $\ket{000}$, and the rotation gate is defined as $R_{\rm YX}(\theta):=\exp(-i\sigma_Y^{\mathsf{A}}\sigma_X^{\mathsf{S}}\theta)$ with $\sigma_P^{\mathsf{A}~(\mathsf{S})}$ denoting the Pauli-$P$ operator on the ancilla (system). After the measurement, we use the relation $\braket{\sigma_{Z}^{\mathsf{A}}}=\cos(\omega T'/2)$ to determine the final estimate of $\omega$.
• Figure 3: Experimental demonstration on an NMR platform. In (a) and (b), the relative errors are plotted as functions of the clock uncertainty for the SWAP interaction and the CNOT interaction, respectively. Here the label S (E) denotes the system (environment). We consider both when the clock uncertainty distribution is fixed to be the uniform distribution over $[-\epsilon,\epsilon]$ and when it can be any bounded distribution over this interval. The length of the errorbar indicates the standard error in 10 repetitions of the experiment.
• Figure 4: The detailed control strategy for frequency estimation. The qubit of ancilla, system, and environment are labeled by 0, 1, and 2 respectively. All state is set to be $\ket{0}$ at the beginning. The gates $H,~S,$ and$~X$ in the circuit are basic quantum logic gates, representing Hadamard, $\pi/2$-phase, and Pauli-$X$ gate respectively.
• Figure 5: The observables evolve over time. The observables $O_{\rm FE}$ and $O_{\rm CE}$ serve as estimators of the unknown frequency $\omega$, providing optimal estimation efficiency in FE and CE case, respectively. (a) illustrates the observable in the FE case, which exhibits rapid fluctuations due to the dynamical evolution of the interaction Hamiltonian. The red crosses indicate the points where the Quantum Fisher Information (QFI) reaches its maximum, corresponding to the value of $\cos(\omega T /2)$. (b) depicts the controlled evolution.

Theorems & Definitions (5)

• Theorem 1
• Corollary 1
• proof
• proof
• proof