Contextuality and Wigner-function negativity in qubit quantum computation

We describe schemes of quantum computation with magic states on qubits for which contextuality and negativity of the Wigner function are necessary resources possessed by the magic states. These schemes satisfy a constraint. Namely, the non-negativity of Wigner functions must be preserved under all available measurement operations. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.

Figure 1

Schematic view of the $n$-qubit state space. The disk represents the space of proper quantum states, the square shows the states describable by a noncontextual hidden variable model, and the lower triangle indicates the states with non-negative Wigner function.

Figure 2

Peres-Mermin square. For the restriction to CSS-ness-preserving operations, the six observables in the top two rows are in the set $\mathcal{O}$ while the observables in the bottom row are in $M$ but not in $\mathcal{O}$. In rebit QCSI, they can be measured individually but not jointly. The figure is adapted from Ref. [3].

Figure 3

The measurements of observables $O_i^{\prime }$ are propagated backwards in time to act on the initial state, by conjugation under the interspersed unitaries. Since only the measurement statistics is of interest, the trailing unitaries may be removed from the resulting circuit.

Figure 4

State space for the one-qubit states $\rho =(I+\mathbf{r}\vec{\sigma })/2$. The physical states lie within or on the Bloch sphere (BS). The two tetrahedra contain the states positively represented by the Wigner functions $W$ and $\overline{W}$, respectively. The state space describable by a noncontextual HVM is a cube with corners $(\pm 1,\pm 1,\pm 1)$; also see Ref. [47]. It contains the Bloch ball.

Figure 5

Relation between Algorithms 1 and 2. Sampling from the probability distribution underlying a noncontextual HVM may be viewed as a two-stage process. Stage 1: Sampling from a probability distribution $\left\{p_i\right\}$ over Wigner functions, Stage 2: Sampling from the phase space with respect to the Wigner function chosen in the first stage.

Figure 6

QCSI with modified cluster state of Eqs. (44a) and (44b) as magic state, which is subjected to measurements of local Pauli operators $X_i, Y_j, Z_k$, for all $i,j\ne i,k\ne i,j\in \mathcal{V}$. The role of the $Z$ measurements is to cut out of the plane a web corresponding to some layout of a quantum circuit, and the $X$ measurements drive the MBQC simulation of this circuit [48]. The qubit in $\mathcal{R}$ is displayed in red. By rerouting a wire piece, one may choose between implementing and not implementing a non-Clifford gate. (a) Identity operation on the logical state space. (b) Logical gate $e^{i\pi /4Z}$.