  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 (CM), Pauli-based computation (PBC) and instantaneous
quantum polynomial time (IQP) circuits.
