Robust qudit Hamiltonian engineering and dynamical decoupling*

We develop a formalism for the Hamiltonian engineering and decoupling in qudit systems beyond spin-1/2. Overcoming the lack of easily visualized geometric structures in the qudit case, we uncover a graph structure that generalizes the qubit Bloch sphere, and identify a choice of pulses that significantly simplifies the incorporation of robustness conditions. Exploiting certain structures in the "decoupling frame graph" that have close analogies with the qubit case, we design a decoupling sequence for strongly disordered, interacting ensembles of spin-1 nitrogen-vacancy centers, in which we experimentally demonstrate an order of magnitude improvement in coherence time over existing sequences. Motivated by recent results on universal decoupling using spherical t-designs, we discuss the outlook toward higher spin sensing, which promises superior sensitivities over their spin-1/2 counterparts.