Arda Aydin

We study relationships between permutation-invariant, bosonic Fock-state, and spin codes, which arise in different physical systems, but exhibit close mathematical affinity. We show that, starting with classical ell-1 codes, it is possible to construct qudit permutationally invariant (PI) codes of arbitrary dimension, spin codes, and Fock state codes, called collectively SU(q) codes. To maintain control of the code parameters in this transition, we rely on a classic result from convex geometry known as Tverberg's theorem. Constructing ell-1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions, we obtain new families of SU(q) codes with distance that scales almost linearly with the code length N. This improves upon the existing designs for all the three code families and yields a conceptually new framework for constructing spin codes. We further present explicit constructions of SU(2) codes with shorter length or lower total spin/excitation than the known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of some short codes with distance larger than the known constructions.