Benchmarking of dynamically corrected gates for the exchange-only spin qubit in $1/f$ 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 $\alpha\gtrsim1.5$, the dynamically corrected gates offer considerable reduction in the gate error and such 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 $\alpha\lesssim1.5$. This critical value $\alpha_c\approx1.5$ is comparatively 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.