Discrete Wigner functions and quantum computational speedup

Abstract

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions $W$ to represent quantum states in a finite Hilbert space dimension $d$. I characterize the set ${C}_{d}$ of states having non-negative $W$ simultaneously in all definitions of $W$ in this class. For $d\ensuremath{\leqslant}5$ I show ${C}_{d}$ is the convex hull of stabilizer states. This supports the conjecture that negativity of $W$ is necessary for exponential speedup in pure-state quantum computation.