Noise-compensating pulses for electrostatically controlled silicon spin qubits

We study the performance of supcode—a family of dynamically correcting pulses designed to cancel simultaneously both Overhauser and charge noise for singlet-triplet spin qubits—adapted to silicon devices with electrostatic control. We consider both natural Si and isotope-enriched Si systems, and in each case we investigate the behavior of individual gates under static noise and perform randomized benchmarking to obtain the average gate error under realistic $1/f$ noise. We find that in most cases supcode pulses offer roughly an order of magnitude reduction in gate error, and especially in the case of isotope-enriched Si, supcode yields gate operations of very high fidelity. We also develop a version of supcode that cancels the charge noise only, “$\delta J$-supcode,” which is particularly beneficial for isotope-enriched Si devices where charge noise dominates Overhauser noise, offering a level of error reduction comparable to the original supcode while yielding gate times that are $30\%–50\%$ shorter. Our results show that the supcode noise-compensating pulses provide a fast, simple, and effective approach to error suppression, bringing gate errors well below the quantum error correction threshold in principle.

Figure 1

Pulse shape for the Hadamard gate $R(\widehat{x}+\widehat{z},\pi )$. (a) Full supcode canceling both Overhauser and charge noise. (b) $\delta J$-supcode canceling charge noise only.

Figure 2

Infidelity of the Hadamard gate vs error in detuning (charge noise) $\delta \epsilon /{\epsilon }_0$ for (a) $\delta h/h=3\%$ relevant to natural Si and (b) $\delta h/h=2\times {10}^{-5}$ relevant to isotope-enriched Si. Infidelity is shown for the naive pulse (black solid), full supcode (red dashed), and $\delta J$-supcode (blue dotted). Note the different scales of the $y$ axis.

Figure 3

Decay of the gate fidelity via randomized benchmarking under $1/f$ noise for charge noise with amplitude ${\beta }_{\epsilon }/{\epsilon }_0=1\%$ for (a) ${\beta }_h/h=3\%$ relevant to natural Si and (b) ${\beta }_h/h=2\times {10}^{-5}$ relevant to isotope-enriched Si. The naive pulse, full supcode, and $\delta J$-supcode are shown as black, red, and blue lines, respectively, and the thin black dotted-dashed, red dashed, and blue dotted lines are fits of the corresponding data to $\frac{1}{2}(1+e^{-\gamma n})$. Note the different scales of both the $x$ and $y$ axes.

Figure 4

Average error per gate vs amplitude of the charge noise ${\beta }_{\epsilon }/{\epsilon }_0$ for (a) ${\beta }_h/h=3\%$ relevant to natural Si and (b) ${\beta }_h/h=2\times {10}^{-5}$ relevant to isotope-enriched Si. The gate error is shown for the naive pulses (black solid), full supcode (red dashed), and $\delta J$-supcode (blue dotted). Note the different scales of the $y$ axis.

Figure 5

The supcode improvement ratio vs charge noise exponent $\gamma$ when nuclear noise is absent. The result of full supcode is shown as the red dashed line and that of $\delta J$-supcode is shown as the blue dotted line. The yellow area shows the regime where supcode reduces error.