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 $1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$, where the exponent $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$, 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 $\ensuremath{\alpha}\ensuremath{\lesssim}1.5$. This critical value of ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.5$ 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.
Geometric quantum gates are performed by using the geometric phase, making them particularly robust to the pulse amplitude error. However, in many systems, such as the silicon-based spin qubits, the off-resonance error is the dominant noise, which can cause dephasing and is always difficult to deal with for a geometric gate. Thus how to combat with the off-resonance error is very significant for the application of the geometric gates. A recent work [S. Li and Z.-Y. Xue, Phys. Rev. Appl. 16, 044005 (2021).] revealed that by inserting two $\ensuremath{\pi}$ pulses into the evolution paths, the holonomic quantum gate is effective to suppress the pulse amplitude error, however, it is still useless for combating the off-resonance error. Inspired by this work, we combine using the techniques of dynamical correction and path design. Surprisingly, we find that, by picking up a specific evolution path inserted by only a $\ensuremath{\pi}$-pulse, the obtained optimized geometric gate is robust to the off-resonance error, assuming the noise is static. Further, by calculating the filter function considering the realistic
1/f-type noise in silicon, the related results show that the performance of the optimized geometric gate can also surpass both the conventional geometric gate and the naive dynamical gate constructed without using the geometric phase. Our results indicate dynamical correction is a powerful tool to improve the geometric quantum computation to achieve a high-fidelity quantum gate in silicon.
We study the performance of composite pulses in the presence of time-varying control noise on a single qubit. These protocols, originally devised only to correct for static, systematic errors, are shown to be robust to time-dependent non-Markovian noise in the control field up to frequencies as high as $\ensuremath{\sim}10%$ of the Rabi frequency. Our study combines a generalized filter-function approach with asymptotic dc-limit calculations to give a simple analytic framework for error analysis applied to a number of composite-pulse sequences relevant to nuclear magnetic resonance as well as quantum information experiments. Results include examination of recently introduced concatenated composite pulses and dynamically corrected gates, demonstrating equivalent first-order suppression of time-dependent fluctuations in amplitude and/or detuning, as appropriate for the sequence in question. Our analytic results agree well with numerical simulations for realistic
1/f noise spectra with a roll-off to
1/f2, providing independent validation of our theoretical insights.