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We investigate the problem of evaluating the output probabilities of Cliffordcircuits with nonstabilizer product input states. First, we consider the casewhen the input state is mixed, and give an efficient classical algorithm toapproximate the output probabilities, with respect to the $l_1$ norm, of alarge fraction of Clifford circuits. The running time of our algorithmdecreases as the inputs become more mixed. Second, we consider the case whenthe input state is a pure nonstabilizer product state, and show that a similarefficient algorithm exists to approximate the output probabilities, when asuitable restriction is placed on the number of qubits measured. Thisrestriction depends on a magic monotone that we call the Pauli rank. We applyour results to give an efficient output probability approximation algorithm forsome restricted quantum computation models, such as Clifford circuits withsolely magic state inputs (CM), Pauli-based computation (PBC) and instantaneousquantum polynomial time (IQP) circuits.

Efficient classical simulation of Clifford circuits with nonstabilizer input states