Efficient classical simulation of Clifford circuits with nonstabilizer input states

We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to approximate the output probabilities, with respect to the l_{1} norm, of a large fraction of Clifford circuits. The running time of our algorithm decreases as the inputs become more mixed. Second, we consider the case when the input state is a pure nonstabilizer product state, and show that a similar efficient algorithm exists to approximate the output probabilities, when a suitable restriction is placed on the number of qubits measured. This restriction depends on a magic monotone that we call the Pauli rank. We apply our results to give an efficient output probability approximation algorithm for some restricted quantum computation models, such as Clifford circuits with solely magic state inputs, Pauli-based computation, and instantaneous quantum polynomial time circuits. Finally, we discuss the relationship between Pauli rank and stabilizer rank.