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We show that qubit stabilizer states can be represented by non-negativequasi-probability distributions associated with a Wigner-Weyl-Moyal formalismwhere Clifford gates are positive state-independent maps. This is accomplishedby generalizing the Wigner-Weyl-Moyal formalism to three generators instead oftwo---producing an exterior, or Grassmann, algebra---which results in Cliffordgroup gates for qubits that act as a permutation on the finite Weyl phase spacepoints naturally associated with stabilizer states. As a result, a non-negativeprobability distribution can be associated with each stabilizer state'sthree-generator Wigner function, and these distributions evolvedeterministically to one another under Clifford gates. This corresponds to ahidden variable theory that is non-contextual and local for qubit Cliffordgates while Clifford (Pauli) measurements have a context-dependentrepresentation. Equivalently, we show that qubit Clifford gates can beexpressed as propagators within the three-generator Wigner-Weyl-Moyal formalismwhose semiclassical expansion is truncated at order $\hbar^0$ with a finitenumber of terms. The $T$-gate, which extends the Clifford gate set to onecapable of universal quantum computation, require a semiclassical expansion ofthe propagator to order $\hbar^1$. We compare this approach to previousquasi-probability descriptions of qubits that relied on the two-generatorWigner-Weyl-Moyal formalism and find that the two-generator Weyl symbols ofstabilizer states result in a description of evolution under Clifford gatesthat is state-dependent, in contrast to the three-generator formalism. We havethus extended Wigner non-negative quasi-probability distributions from the odd$d$-dimensional case to $d=2$ qubits, which describe the non-contextuality ofClifford gates and contextuality of Pauli measurements on qubit stabilizerstates.

Discrete Wigner Formalism for Qubits and Non-Contextuality of Clifford Gates on Qubit Stabilizer States