Benchmarking of dynamically corrected gates for the exchange-only spin qubit in a $1/f$ noise environment

We study theoretically the responses of the dynamically corrected gates to time-dependent noises in the exchange-only spin-qubit system. We consider $1/f$ noises having spectra proportional to $1/{\omega }^{\alpha }$, where the exponent $\alpha$ indicates the strength of correlation within the noise. The quantum gate errors due to noises are extracted from a numerical simulation of randomized benchmarking and are compared between the application of uncorrected operations and that of dynamically corrected gates robust against the hyperfine noise. We have found that for $\alpha \gtrsim 1.5$, the dynamically corrected gates offer a considerable reduction in the gate error and such a reduction is approximately two orders of magnitude for the experimentally relevant noise exponent. On the other hand, no improvement of the gate fidelity is provided for $\alpha \lesssim 1.5$. This critical value of ${\alpha }_c\approx 1.5$ is larger than that for the cases for the singlet-triplet qubits. The filter transfer functions corresponding to the dynamically corrected gates are also computed and compared to those derived from uncorrected pulses. Our results suggest that the dynamically corrected gates are useful measures to suppress the hyperfine noise when operating the exchange-only qubits.

Figure 1

(a) Schematic of an exchange-only qubit with exchange interactions between adjacent spins indicated. (b) Example of the dynamically corrected gates achieving a rotation of $R_{12}(\pi /2)$, canceling the leading-order contribution from the hyperfine noise. Here $t_0$ is an arbitrary time unit and $J_{12}{t}_0$ and $J_{23}{t}_0$ are dimensionless control fields. The uncorrected pulse is shown as the black line, while the hyperfine-noise-corrected pulse sequences are shown as red (gray) lines.

Figure 2

Example of the $1/f$ noise used in the simulation. (a) Noise as a function of time. (b) Power spectral density corresponding to (a), $S\left(\omega \right)=A/{\left(\omega t_0\right)}^{1.5}$. The noise amplitude $A$ has been scaled such that $S(\omega =1/t_0)\approx 1/t_0$. Here both the $x$- and $y$-axis labels are shown as dimensionless quantities.

Figure 3

Randomized benchmarking of single-qubit Clifford gates undergoing $1/f$ noises with different noise exponents $\alpha$ as indicated. Here $n$ is the number of gates and $F$ indicates the fidelity. The results for uncorrected and corrected gates are shown as black and red (gray) lines, respectively. The dimensionless noise amplitudes for each panel are (a) $A{t}_0={10}^{-3}$, (b) ${10}^{-3}$, (c) ${10}^{-3.5}$, (d) ${10}^{-4}$, (e) ${10}^{-5}$, and (f) ${10}^{-6}$. The exponential fits according to Eq. (11) are shown as thin dashed lines with the same color of the data. Note that for uncorrected gates with $1.5\le \alpha \le 3$, the data deviate from the exponential behavior. For other curves, the data and fitting lines overlap, indicating a good exponential fit.

Figure 4

Fidelity decay constant $\gamma$ vs dimensionless noise amplitude $A{t}_0$ for $1/f$ noises with different noise exponents $\alpha$ as indicated. The results for uncorrected and corrected gates are shown as black and red (gray) lines, respectively.

Figure 5

Improvement ratio $\kappa$ of the dynamically corrected gates vs the noise exponent $\alpha$. The dashed line marks $\kappa =1$.

Figure 6

Filter transfer functions of the Hadamard gate $R(\widehat{x}+\widehat{z},\pi )$. The filter transfer functions for the uncorrected rotation are shown as black lines, while those for corrected rotations are shown as red (gray) lines. (a)–(d) Components of the filter transfer functions $Q_{11}, Q_{33}, Q_{44}$, and $Q_{66}$. (e) Total filter transfer function for the noise ${\mathrm{\Delta }}_A$ calculated from Eq. (17). (f) Total filter transfer function for the noise ${\mathrm{\Delta }}_B$ calculated from Eq. (18).