Simulation of quantum computation with magic states via Jordan-Wigner transformations

Negativity in certain quasiprobability representations is a necessary condition for a quantum computational advantage. Here we define a quasiprobability representation exhibiting this property with respect to quantum computations in the magic state model. It is based on generalized Jordan-Wigner transformations, and it has a close connection to the probability representation of universal quantum computation based on the $\mathrm{\ensuremath{\Lambda}}$ polytopes. For each number of qubits, it defines a polytope contained in the $\mathrm{\ensuremath{\Lambda}}$ polytope with some shared vertices. It leads to an efficient classical simulation algorithm for magic state quantum circuits for which the input state is positively represented, and it outperforms previous representations in terms of the states that can be positively represented.