States with maximal magic

The notion of a “magic state” for Clifford-gate quantum computation was introduced by Bravyi and Kitaev in 2004. Broadly speaking, magic is a resource that allows Clifford-gate quantum computers to become universal, enabling them to run any quantum algorithm, such as Shor's factorization. Yet, quantifying magic in terms of a precise numerical scale remains an open area of research. In Phys. Rev. Lett. 128, 050402 (2022), Leone, Oliviero, and Hamma provided a computable measure of magic for qubit quantum computers called stabilizer entropy. In this talk, I present a generalization of their notion to qudit systems (systems with dimension d≥2), showing that is maximized uniquely by the states of the Weyl-Heisenberg covariant Symmetric Informationally Complete quantum measurements (SIC-POVMs). Thus, SIC-POVM states obtain a unique nonclassical character with respect to this measure of magic and perhaps for quantum computation as well. This is surprising as the initial motivation for studying SIC-POVMs had nothing to do with quantum computation, being strongly rooted instead in quantum foundations. Also, it reveals an intimate relationship between the existence of states with maximal magic and the SIC-POVM existence problem, that Appleby et al. showed to be deeply linked to Hilbert’s 12^th problem in algebraic number theory. This suggests that maximal magic with respect to stabilizer entropy is a rather robust notion.