  By the Gottesman-Knill Theorem, the outcome probabilities of Clifford
circuits can be computed efficiently. We present an alternative proof of this
result for quopit Clifford circuits (i.e., Clifford circuits on collections of
$p$-level systems, where $p$ is an odd prime) using Feynman's sum-over-paths
technique, which allows the amplitudes of arbitrary quantum circuits to be
expressed in terms of a weighted sum over computational paths. For a general
quantum circuit, the sum over paths contains an exponential number of terms,
and no efficient classical algorithm is known that can compute the sum. For
quopit Clifford circuits, however, we show that the sum over paths takes a
special form: it can be expressed as a product of Weil sums with quadratic
polynomials, which can be computed efficiently. This provides a method for
computing the outcome probabilities and amplitudes of such circuits
efficiently, and is an application of the circuit-polynomial correspondence
which relates quantum circuits to low-degree polynomials.
