Randomized Benchmarking of Barrier versus Tilt Control of a Singlet-Triplet Qubit

Decoherence due to charge noise is one of the central challenges in using spin qubits in semiconductor quantum dots as a platform for quantum information processing. Recently, it has been experimentally demonstrated in both Si and GaAs singlet-triplet qubits that the effects of charge noise can be suppressed if qubit operations are implemented using symmetric barrier control instead of the standard tilt control. Here, we investigate the key issue of whether the benefits of barrier control persist over the entire set of single-qubit gates by performing randomized benchmarking simulations. We find the surprising result that the improvement afforded by barrier control depends sensitively on the amount of spin noise: for the minimal nuclear spin noise levels present in Si, the coherence time improves by more than 2 orders of magnitude whereas in GaAs, by contrast the coherence time is essentially the same for barrier and tilt control. However, we establish that barrier control becomes beneficial if qubit operations are performed using a new family of composite pulses that reduce gate times by up to 90%. With these optimized pulses, barrier control is the best way to achieve high-fidelity quantum gates in singlet-triplet qubits.

Figure 1

Pulse profiles of the unoptimized (red/gray lines) and optimized (black lines) pulse sequences for (a) $R(\widehat{z},\pi )$ and (b) $R(\widehat{x}+\widehat{y}+\widehat{z},2\pi /3)$. Here, $t_0$ is a time scale that is taken to be $1/h$ throughout this paper.

Figure 2

Results of randomized benchmarking with Gaussian noise (i.e., the quasistatic limit) for unoptimized (red or gray) and optimized (black) pulses. The dashed lines are exponential fits as detailed in the main text. The magnetic field gradient and nuclear spin noise values are $h=23\mathrm{MHz}$ and $\delta h=0$ for (a) and (b), $h=40\mathrm{MHz}$ and ${\sigma }_h=11.5\mathrm{MHz}$ for (c) and (d), and $h=40\mathrm{MHz}$ and ${\sigma }_h=23\mathrm{MHz}$ for (e) and (f). For the left column [(a), (c), (e)], we use barrier control, and thus we take ${\sigma }_J=0.00426J$, while we use tilt control in the right column [(b), (d), (f)] and thus we use ${\sigma }_J=0.0563J$. We indicate the fitted $T_2$ values on this figure in the same color as the lines that they correspond to, and we provide the fitting parameters in the Supplemental Material [44].

Figure 3

Randomized benchmarking results of the optimized pulse sequences under $1/f^{2.6}$ nuclear noise and $1/f^{0.7}$ charge noise (i.e. ${\alpha }_h=2.6$ and ${\alpha }_J=0.7$). The solid curves are our numerical results and the dashed curves are fits. (a) and (b) correspond to $h=23\mathrm{MHz}$ and $\delta h=0$, while (c) and (d) correspond to $h=40\mathrm{MHz}$ and $\delta h\ne 0$. We use barrier control in (a) and (c), and tilt control in (b) and (d). We indicate the fitted $T_2$ values in the figures, and provide other parameters in the Supplemental Material [44].

Figure 4

$T_2$ vs $\alpha$ for optimized pulse sequences. Solid lines show the results for barrier control, dashed lines for tilt control. Black lines: nuclear spin noise not present and ${\alpha }_J=\alpha$ for charge noise. Red lines: ${\alpha }_h={\alpha }_J=\alpha$. Blue lines: ${\alpha }_h=2.6$ and ${\alpha }_J=\alpha$.