One of the most common way to obtain universality in quantum computation
is by the injection of magic states. This raises the question:
Which quantum properties of these states are responsible for the
gain in computational power?
Wigner functions negativity and contextuality have recently been
proposed to explain this extra power for qupits (p-level systems for
odd p). Unfortunately the case of qubits seems much more involved.
In this talk, I will recall the construction of Discrete Wigner
functions and their relation with contextuality and quantum
computation for qupits. Then I will consider the case of real
2-level systems and I will explain how to resurrect most
of the previous results. This is a first step toward qubits.
Based on joint work with Philippe Allard Guerin, Jacob Bian and Robert Raussendorf.