  Negativity in certain quasiprobability representations is a necessary
condition for a quantum computational advantage. Here we define a new
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 $\Lambda$
polytopes. For each number of qubits it defines a polytope contained in the
$\Lambda$ 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.
