Hidden Variable Model for Universal Quantum Computation with Magic States on Qubits

We show that every quantum computation can be described by a probabilistic update of a probability distribution on a finite phase space. Negativity in a quasiprobability function is not required in states or operations. Our result is consistent with Gleason’s theorem and the Pusey-Barrett-Rudolph theorem.

Figure 1

One-qubit model. (a) The state space ${\mathrm{\Lambda }}_1$ is a cube with eight vertices corresponding to the phase point operators $A_{\alpha }=[I+(-1{)}^{s_x}X+(-1{)}^{s_y}Y+(-1{)}^{s_z}Z]/2$, with $\alpha =(s_x,s_y,s_z)\in {\mathbb{Z}}_2^3$. The physical one-qubit states lie on or in the Bloch sphere that is contained in ${\mathrm{\Lambda }}_1$ and touches the boundary of ${\mathrm{\Lambda }}_1$ at six points corresponding to the six one-qubit stabilizer states. (b) Update of the phase point operators $A_{\alpha }$ under measurement of the Pauli observable $Z$. Each red arrow represents a transition probability of $1/2$.