# Codes and Expansions (CodEx) Seminar

## Steven Flammia (AWS Center for Quantum Computing)Ghosts, Necromancy, and Zauner's Conjecture

In 1999, Zauner conjectured that a maximal set of $d^2$ equiangular lines always exists in $\mathbb{C}^d$. Such sets are called SICs and are known to exist for only finitely many $d$. The strongest form of Zauner's conjecture states the specific minimal field extensions over $\mathbb{Q}$ where the SICs are defined. I will review this field structure, and show how we are naturally led to consider certain objects dual to a SIC that we call ghosts. Ghosts have many interesting geometric and number theoretic properties, and may be more amenable to direct construction and analysis than SICs. I will describe a procedure that we call necromancy which conjecturally reconstructs a dual SIC from a (numerical approximation of) a ghost. Using necromancy, we reanimate the ghosts in $d=100$ and construct four (numerical) SICs, three of which are new. We conjecture that every Weyl-Heisenberg covariant SIC in $d>3$ dimensions can be obtained in this way. This is joint work with Marcus Appleby and Gene Kopp.