Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states
Abstract
We show that qubit stabilizer states can be represented by nonnegative quasiprobability distributions associated with a WignerWeylMoyal formalism where Clifford gates are positive stateindependent maps. This is accomplished by generalizing the WignerWeylMoyal formalism to three generators instead of two—producing an exterior, or Grassmann, algebra—which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a nonnegative probability distribution can be associated with each stabilizer state's threegenerator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is noncontextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a contextdependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the threegenerator WignerWeylMoyal formalism whose semiclassical expansion is truncated at order ℏ^{0} with a finite number of terms. The T gate, which extends the Clifford gate set to one capable of universal quantum computation, requires a semiclassical expansion of the propagator to order ℏ^{1}. We compare this approach to previous quasiprobability descriptions of qubits that relied on the twogenerator WignerWeylMoyal formalism and find that the twogenerator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is statedependent, in contrast to the threegenerator formalism. We have thus extended Wigner nonnegative quasiprobability distributions from the odd d dimensional case to d =2 qubits, which describe the noncontextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.
 Publication:

Physical Review A
 Pub Date:
 December 2017
 DOI:
 10.1103/PhysRevA.96.062134
 arXiv:
 arXiv:1705.08869
 Bibcode:
 2017PhRvA..96f2134K
 Keywords:

 Quantum Physics
 EPrint:
 Phys. Rev. A 96, 062134 (2017)