  The implementation of physical symmetries into problem descriptions allows
for the reduction of parameters and computational complexity. We show the
integration of the permutation symmetry as the most restrictive discrete
symmetry into quantum circuits. The permutation symmetry is the supergroup of
all other discrete groups. We identify the permutation with a
$\operatorname{SWAP}$ operation on the qubits. Based on the extension of the
symmetry into the corresponding Lie algebra, quantum circuit element
construction is shown via exponentiation. This allows for ready integration of
the permutation group symmetry into quantum circuit ansatzes. The scaling of
the number of parameters is found to be $\mathcal{O}(n^3)$, significantly lower
than the general case and an indication that symmetry restricts the
applicability of quantum computing. We also show how to adapt existing circuits
to be invariant under a permutation symmetry by modification.
