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We study theoretically the responses of the dynamically corrected gates totime-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 quantumgate errors due to noises are extracted from a numerical simulation ofRandomized Benchmarking, and are compared between the application ofuncorrected operations and that of dynamically corrected gates robust againstthe hyperfine noise. We have found that for $\alpha\gtrsim1.5$, the dynamicallycorrected gates offer considerable reduction in the gate error and suchreduction is approximately two orders of magnitude for the experimentallyrelevant noise exponent. On the other hand, no improvement of the gate fidelityis 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 gatesare also computed and compared to those derived from uncorrected pulses. Ourresults suggest that the dynamically corrected gates are useful measures tosuppress the hyperfine noise when operating the exchange-only qubits.

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