  Permutational Quantum Computing (PQC) [\emph{Quantum~Info.~Comput.},
\textbf{10}, 470--497, (2010)] is a natural quantum computational model
conjectured to capture non-classical aspects of quantum computation. An
argument backing this conjecture was the observation that there was no
efficient classical algorithm for estimation of matrix elements of the $S_n$
irreducible representation matrices in the Young's orthogonal form, which
correspond to transition amplitudes of a broad class of PQC circuits. This
problem can be solved with a PQC machine in polynomial time, but no efficient
classical algorithm for the problem was previously known. Here we give a
classical algorithm that efficiently approximates the transition amplitudes up
to polynomial additive precision and hence solves this problem. We further
extend our discussion to show that transition amplitudes of a broader class of
quantum circuits -- the Quantum Schur Sampling circuits -- can be also
efficiently estimated classically.
