Robust qudit hamiltonian engineering with spherical 2(d-1)-designs

Reshaping a native interaction into a desired form via pulsed coherent control, so-called Hamiltonian engineering, is a ubiquitous technique in quantum science. In this work, we provide a group-theoretic classification for conditions a pulse-sequence must satisfy in order to transform a native qudit interaction into a one with a desired continuous symmetry, which can be one of the many continuous subgroups of $SU(d)$. We find that spherical $2(d-1)$-designs associated to suitably generalized Bloch spheres can be used to construct universal and experimentally robust pulse sequences required to engineer these symmetries. Our approach offers an efficient method for quantum simulation with global control, opening the door to near-term applications ranging from high-spin entanglement enhanced sensing to quantum simulation of non-abelian lattice gauge theories.