Filter design for hybrid spin gates

The impact of control sequences on the environmental coupling of a quantum system can be described in terms of a filter. Here we analyze how the coherent evolution of two interacting spins subject to periodic control pulses, using the example of a nitrogen vacancy center coupled to a nuclear spin, can be described in the filter framework in both the weak- and the strong-coupling limit. A universal functional dependence around the filter resonances then allows for tuning the coupling type and strength. We show how the validity range of the filter description, originally limited to small rotation angles, can be extended to the long time limit by time-sliced evolution sequences. Based on that insight, the construction of tunable, noise decoupled, conditional gates composed of alternating pulse sequences is proposed. In particular such an approach can lead to a significant improvement in fidelity as compared to a strictly periodic control sequence. Moreover we analyze the decoherence impact, the relation to the filter for classical noise known from dynamical decoupling sequences, and we outline how an alternating sequence can improve spin sensing protocols.