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.