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Dynamical decoupling (DD) is a promising tool for preserving the quantumstates of qubits. However, small imperfections in the control pulses canseriously affect the fidelity of decoupling, and qualitatively change theevolution of the controlled system at long times. Using both analytical andnumerical tools, we theoretically investigate the effect of the pulse errorsaccumulation for two aperiodic DD sequences, the Uhrig's DD UDD) protocol [G.S. Uhrig, Phys. Rev. Lett. {\bf 98}, 100504 (2007)], and the Quadratic DD (QDD)protocol [J. R. West, B. H. Fong and D. A. Lidar, Phys. Rev. Lett {\bf 104},130501 (2010)]. We consider the implementation of these sequences using theelectron spins of phosphorus donors in silicon, where DD sequences are appliedto suppress dephasing of the donor spins. The dependence of the decouplingfidelity on different initial states of the spins is the focus of our study. Weinvestigate in detail the initial drop in the DD fidelity, and its long-termsaturation. We also demonstrate that by applying the control pulses alongdifferent directions, the performance of QDD protocols can be noticeablyimproved, and explain the reason of such an improvement. Our results can beuseful for future implementations of the aperiodic decoupling protocols, andfor better understanding of the impact of errors on quantum control of spins.

Aperiodic dynamical decoupling sequences in presence of pulse errors