Efficiency of optimal control for noisy spin qubits in diamond

TL;DR

This work analyzes decoherence in NV-center spin qubits and how the environmental noise correlation time $\tau_\delta$ shapes optimal control pulses for spin inversion. Using the dCRAB quantum optimal control method under Ornstein-Uhlenbeck noise with detuning $\delta(t)$ and amplitude error $\epsilon(t)$, the authors build a control landscape and quantify performance via averaged fidelity over noise realizations, validating numerically and experimentally. They explore how optimization options (degrees of freedom, basis size $N_c$, and wiggles via $\beta_{\max}$) affect pulse shapes and robustness, finding that pulse features adapt with $\tau_\delta$ and that experimental feasibility imposes meaningful constraints on the optimized control. The study provides practical guidelines for implementing robust optimal control in noisy solid-state qubits, informing pulse design for NV centers and similar quantum platforms under realistic noise.

Abstract

Decoherence is a major challenge for quantum technologies. A way to mitigate its negative impact is by employing quantum optimal control. The decoherence dynamics varies significantly based on the characteristics of the surrounding environment of qubits, consequently affecting the outcome of the control optimization. In this work, we investigate the dependence of the shape of a spin inversion control pulse on the correlation time of the environment noise. Furthermore, we analyze the effects of constraints and optimization options on the optimization outcome and identify a set of strategies that improve the optimization performance. Finally, we present an experimental realization of the numerically-optimized pulses validating the optimization feasibility. Our work serves as a generic yet essential guide to implementing optimal control in the presence of realistic noise, e.g., in nitrogen-vacancy centers in diamond.

Figures (9)

• Figure 1: Single random $\delta$ noise samples (top) and the corresponding Rabi oscillation of the sample-averaged population inversion represented by $\overline{\left\langle{\sigma_z}\right\rangle}$ (bottom). They are obtained by fixing the coherence decay time, $T_2^*=0.1\ \mathrm{\mu s}$, and by varying the correlation time $\tau_\delta$ of the $\delta$ noise. We consider no amplitude noise in this example simulation, i.e., $\epsilon=0$. One Rabi period is $2\pi/\Omega_1$, where $\Omega_1=2\pi\,1\ \mathrm{MHz}$. The inset in the bottom figure shows small fluctuations in the average value due to the finite sample size represented by the standard deviation of the fluctuation statistics. It is obtained by repeating the simulation with different noise realizations with $N_\mathrm{sample}=1500, N_\mathrm{rep}=100$. The values for $\tau_\delta$ are from Table \ref{['Table:noise_spectra']}.
• Figure 2: Optimized modulation components $f_x(t)=f(t)\cos[\phi(t)]$ and $f_y(t)=f(t)\sin[\phi(t)]$ vs. the evolution time $t$, for different correlation times of the noise in the detuning $\tau_\delta$. We consider the case where $f(t)$ is optimized with $\phi(t)=\pi/2$, $N_c=5$, and $\beta_\mathrm{max}=3$ for all plots in the figure. On top of each plot, the value of $\tau_\delta$, the optimized cost function $J_\mathrm{opt}$, and the cost function $J_\mathrm{narrow}$ for a short rectangular pulse, which we label narrow, are shown. The black dotted lines show the limits in the magnitude of the optimized function. The fixed parameters used here are $\Omega_1=2\pi\ \mathrm{MHz}, N_\mathrm{iter}=2000, T=0.5\ \mathrm{\mu s}, N_\mathrm{sample} = 1500, N_\mathrm{rep}=100$. These figures are located along the first row of Fig. \ref{['fig_main']} (coded as 1.a.i.)
• Figure 3: Shape of the optimized pulses vs. the evolution time $t$ for correlation time $\tau_\delta=100\,\mu$s exhibiting how the shape of the optimized pulse and the cost function change as the optimization options are modified. The optimization options parameters$[\phi(t), N_c, \beta_\mathrm{max}]$ take a set of values, as follows: (a) 1.a.i. $[\pi/2, 5, 3]$, (b) 2.a.i. $[\mathrm{const.}, 5, 3]$, (c) 3.a.i $[\phi(t), 5, 3]$, (d) 1.b.i. $[\pi/2, 10, 3]$, (e) 1.a.ii. $[\pi/2, 5, 8]$. The components in (b) have the same phase relation for all $t$. The features of the plots are noticeably similar to Fig. \ref{['fig2']}.
• Figure 4: Comparison of experimentally and numerically generated optimized pulse shapes $f_y(t)$ ($f_x(t)=0$ for these cases) vs. evolution time $t$ for selected cases with $\phi=\pi/2$ and $N_c=5$: (a) $\tau_\delta=100, \beta_\mathrm{max}=3$ (case 1.a.i.); (b) $\tau_\delta=100, \beta_\mathrm{max}=8$ (case 1.a.ii); (c) $\tau_\delta=0.01, \beta_\mathrm{max}=3$ (case 1.a.i.); (d) $\tau_\delta=0.01, \beta_\mathrm{max}=8$ (case 1.a.ii.). The post-measurement treatment of the experimental pulse is discussed in the main text. On top of each plot on the left column is the mean absolute error (MAE) of the two pulses.
• Figure 5: Evolution of $\overline{\left\langle{\sigma_z}\right\rangle}$ in the presence of different noise sources--- $\delta$ noise only, $\epsilon$ noise only, and both---for rectangular control pulses under different conditions: (a) short duration, small rotation phase; (b) short duration, large rotation phase; (c) long duration, small rotation phase; (d) long duration, long rotation phase. The value $\phi(t)=\pi/2$ is arbitrarily chosen and is the same for all simulations. The OU parameters of the $\delta$ noise, $\tau_\delta=0.1\ \mathrm{\mu s}$ and $T_2^*=0.1\ \mathrm{\mu s}$, are arbitrarily chosen such that the chosen pulse lengths and rotation phases exhibit different effects. Simulated with $N_\mathrm{sample}=1500,N_\mathrm{rep}=100$.