We consider a system with general decoherence and a quadratic dynamical decoupling sequence (QDD) for the coherence control of a qubit coupled to a bath of spins. We investigate the influence of the geometry and of the initial conditions of the bath on the performance of the sequence. The overall performance is quantified by a distance norm
d. It is expected that
d scales with $\ensuremath{\tau}$, the total duration of the sequence, as ${\ensuremath{\tau}}^{\mathrm{min}{{N}_{x},{N}_{z}}+1}$, where
Nx and
Nz are the number of pulses of the outer and of the inner sequence, respectively. We show both numerically and analytically that the state of the bath can boost the performance of QDD under certain conditions: The scaling of QDD for a given number of pulses can be enhanced by a factor of 2 if the bath is prepared in a highly symmetric state and if the system Hamiltonian is SU(2) invariant.
An efficient optimal-control theory based on the Krotov method is introduced for a non-Markovian open quantum system with a time-nonlocal master equation in which the control parameter and the bath correlation function are correlated. This optimal-control method is developed via a quantum dissipation formulation that transforms the time-nonlocal master equation to a set of coupled linear time-local equations of motion in an extended auxiliary Liouville space. As an illustration, the optimal-control method is applied to find the control sequences for high-fidelity
Z gates and identity gates of a qubit embedded in a non-Markovian bath.
Z gates and identity gates with errors less than ${10}^{\ensuremath{-}5}$ for a wide range of bath decoherence parameters can be achieved for the non-Markovian open qubit system with control over only the ${\ensuremath{\sigma}}_{z}$ term. The control-dissipation correlation and the memory effect of the bath are crucial in achieving the high-fidelity gates.