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 non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive stateindependent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal 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 non-negative probability distribution can be associated with each stabilizer state's three-generator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is non-contextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a context-dependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the three-generator Wigner-Weyl-Moyal formalism whose semiclassical expansion is truncated at order 0 with a finite number of terms. The T -gate, which extends the Clifford gate set to one capable of universal quantum computation, require a semiclassical expansion of the propagator to order 1 . We compare this approach to previous quasi-probability descriptions of qubits that relied on the two-generator Wigner-Weyl-Moyal formalism and find that the twogenerator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is state-dependent, in contrast to the three-generator formalism. We have thus extended Wigner non-negative quasi-probability distributions from the odd d-dimensional case to d = 2 qubits, which describe the non-contextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.

Measurement contextuality and Planck’s constant

“…This is certainly not the case for the simulation of qubit stabilizer states. Preparation and evolution of these states is efficient using the Aaronson-Gottesmann algorithm and these operations are also non-contextual for qubits [22]. This is one of the puzzles we examine in the present paper.…”

“…Applying such an approach requires a path integral formulation of discrete quantum systems. We will use the WWM formalism which has been developed for oddd-dimensional qudits [1,5,13,23] and was recently extended to qubits (d=2) [22]. The WWM formalism is particularly useful for finite-dimensional systems because it uses the conjugate degrees of freedom of 'chords' and 'centers' to define Hamiltonian phase space, instead of momentum and position as is the traditional approach for infinite-dimensional continuous Hilbert spaces.…”

“…The WWM formalism was originally developed for continuous infinite-dimensional systems, and has since been extended to finite-dimensional qudits [5,22,23]. This extension differs depending on whether the qudits are odd-dimensional or even-dimensional (d = 2).…”

“…where 'P.B.' stands for the Poisson bracket and x ¶ ¶ ¬ j and x ¶  ¶ j are right and left derivatives, respectively, as defined in [22].…”

“…Contextuality is a quantum resource whose presence in an experiment has been shown to be equivalent to negativity in the associated discrete Wigner functions and Weyl symbols of the states, operations, and measurements that are involved [9,10]. Preparation contextuality [20] has been shown to be equivalent to the necessity of including higher than order ÿ 0 terms in path integral expansions of unitary operators in the WWM formalism for proper computation of propagation [21,22]. As a result, stabilizer state propagation under Clifford gates can be shown to be non-contextual since it does not require higher than ÿ 0 terms within WWM, or equivalently, the Wigner function of stabilizer states are non-negative and the Weyl symbols of Clifford gates are positive maps [5,13,23].…”