Application of Optimal Control to Time-Resolution Protocol for Quantum Sensing

TL;DR

This work formulates time-resolution quantum sensing as an end-to-end optimal-control problem for a two-level sensor with $H = \frac{\omega_0 + \delta \omega}{2} \hat{\sigma}_z + u(t) \hat{\sigma}_x$, aiming to maximize the sensitivity $\eta$ within a fixed interrogation time $\tau$ under a bound $|u(t)| \le u_{max}$. Using OCT, the authors show a critical interrogation time $T^*$ that separates bang-bang from singular control regimes, and introduce a smooth detune protocol that matches or slightly exceeds the reference YX protocol in the short-$\tau$ regime, while revealing Ramsey-like behavior for longer $\tau$. They compare end-to-end performance with QFI-based criteria, highlighting that maximizing QFI alone can misrepresent practical sensing capabilities. The work also discusses practical considerations, including smoothing discontinuities and a potential application to calibrating baseband flux-pulse distortions in superconducting qubits, suggesting that the detune protocol offers a realistic route toward implementing high time-resolution quantum sensing.

Abstract

Time-resolution protocol of quantum sensing aims to measure the fast temporal variation of an external field and demands a high field sensitivity in a short interrogation time $τ$. Since any operation that evolves the quantum state takes time and is counted as part of the interrogation, evaluating the performance of time-resolution protocol requires a complete end-to-end description of the measurement process. In particular, the initial state has to be one of the sensor qubit's eigenstates in the absence of external fields, and the final projective measurements must be performed in the same eigenstate basis. Building upon prior works which proposed limits for time-resolved sensing using a quantum sensor, we apply optimal control theory to optimize the time-resolution protocol. Our analysis indicates that there exists a critical interrogation time $T^*$: when $τ<T^*$ the optimal protocol is purely bang-bang; when $τ>T^*$ the optimal protocol involves a singular control during the interrogation. In the short-$τ$ regime, which is relevant to high time resolution, we propose a ``detune protocol'' that involves only smooth control during the entire interrogation. As the discontinuities of control pose the main obstacles to experimental realization, we expect the presented detune protocol to be practically useful. In the long-$τ$ regime, the optimal protocol closely resembles the Ramsey sequence; protocols based on maximizing Quantum Fisher Information are constructed to highlight the difference between the theoretically optimal and practically implementable measurements. Effective use of the time-resolution protocol requires a setup where the unknown time-domain signal of interest can be identically and repeatedly generated. As a potentially relevant application, we outline the calibration of baseband flux pulse distortion in the control of superconducting qubits.

Figures (8)

• Figure 1: $\eta_\text{opt}/\tau$ for $u_\text{max}=0.1$, 0.2, 0.5. The reference YX protocol using RWA is provided as the reference. The metrological gain of optimal protocol is more significant for short interrogation time, which is the regime relevant to high time resolution. The critical interrogation time $T^*$ depends on $u_\text{max}$ and is about 0.8 $t_\text{QSL}$ ($t_\text{QSL} = \frac{\pi}{u_\text{max}}$) for $u_\text{max}<0.2$ (blue area). (b) For $u_\text{max}=0.2$, ${\tau} = 0.6 \, t_\text{QSL}$, the optimal control is of BB. Optimality conditions are verified. (c) For $u_\text{max}=0.2$, ${\tau} = 1.2 \, t_\text{QSL}$, the optimal control includes a singular portion in the middle. Optimality conditions -- Eq. \ref{['eqn:u^*(t)']} for B control, Eq. \ref{['eqn:singular_control_value']} for S control, and a constant $\mathcal{H}_\text{oc}(t)$ -- are numerically verified.
• Figure 2: Detune protocol within RWA. (a) The difference of detune values between directly optimizing Eq. \ref{['eqn:eta_Detune']} and the approximation of \ref{['eqn:optimal_detune_RWA_approx']}. (b) The resulting $\eta/\tau$ for $u_\text{max}=0.05$ and 0.2. The difference between YX and detune protocols (RWA) is very small. The difference between detune protocols using RWA and the full calculation increases [using the approximate detune of \ref{['eqn:optimal_detune_RWA_approx']}] as $u_\text{max}$ increases.
• Figure 3: Detune protocol in RWA and rotating frame. Trajectories on the Bloch sphere (a) and in $\phi$-$\cos^2 (\theta/2)$ plane (b). In both (a) and (b), the solid blue trajectory corresponds to the no-detune $\Delta \omega = 0$ and zero-field $\delta \omega = 0$ case; dashed blue to no-detune $\Delta \omega = 0$ and finite-field $\delta \omega = 0.1$; solid red to optimal-detune $\Delta \omega = 0.326$ and zero-field $\delta \omega = 0$; dashed red optimal-detune $\Delta \omega = 0.326$ and finite-field $\delta \omega = 0.1$. Within RWA, the detune protocol has a time-independent Hamiltonian and each trajectory corresponds to a rotating axis of the Bloch sphere. Bottom of (a) gives the rotating axes of four trajectories. The $\cos^2(\theta/2)$ is the measurement signal, and the difference at the end of interrogation with and without field quantifies the sensing performance. (b) In $\phi$-$\cos^2 (\theta/2)$ plane, the optimal detune (red) leads to a larger difference compared to the case of no detune (blue). The trajectories are computed using $u_\text{max}=0.2$, $\tau = 0.5 \,t_\text{QSL} = 2.5 \pi$.
• Figure 4: $\eta/\tau$ obtained from the optimal, YX, and detune protocols as a function of $\tau$ ($\tau \in [0, t_\text{QSL}]$) for $u_\text{max}=0.1$ (a) and 0.2 (d) using full calculation. The optimized detune parameters are given in (b) and (e). Notice that the $a$ is either 0 or 0.5. The corresponding optimal, YX, and detune protocol for $\tau = 0.6 \, t_\text{QSL}$ are provided in (c) and (e). The main advantage of the detune protocol is its smoothness over the entire interrogation time.
• Figure 5: Optimal controls that minimize $\mathcal{C}_{\eta^2}$ and $\mathcal{C}_\text{QFI}$ for $u_\text{max} = 0.1$ (a), (b) and $u_\text{max} = 0.5$ (c), (d). The corresponding QFI$/t^2$ and $\langle \hat{\sigma}_z \rangle(t)$ are also present. The interrogation time is $\frac{\tau}{t_\text{QSL}} = 1.2$. The optimal control involves a singular arc in the middle, corresponding to the free evolution in the Ramsey protocol. According to $\langle \hat{\sigma}_z \rangle$, both $\eta$-optimal and QFI-optimal controls bring the quantum sensor to the equator of the Bloch sphere (i.e., $\langle \sigma_z \rangle=0$) as soon as possible. To maximize QFI, the sensor stays on the equator. To maximize $\eta$, the optimal control induces some oscillation around the equator in the last stage of the interrogation.