  In this paper, we exclusively utilize CNOT gates for implementing permutation
groups generated by more than two elements. In Lemma 1, we recall that three
CNOT gates are both necessary and sufficient to execute a two-qubit swap gate
operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT
gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving
forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT
gates is insufficient for implementing a three-qubit swap gate corresponding to
the permutation element (123). Hence six CNOT gates are both necessary and
sufficient for implementing (123). This is done by employing a graph-theoretic
approach to rigorously validate the results in terms of at most five CNOT
gates. Using computational tools, we exhaustively explore all valid circuit
diagrams containing exactly six CNOT gates to successfully execute the swap
gate for (123), by explaining the equivalence classes in Remark 6 and Table 2.
We conclude them in Theorem 7.To extend our analysis to the multiqubit
scenario, we present the reducible and irreducible permutation elements in
Definition 8. We clarify the equivalence between rows in the multi-qubit space
and provide an approximate upper bound for multi-qubits to perform the
aforementioned operations in Theorem 9. The comprehensive exploration of this
paper aims to pave the way for further advancements in understanding quantum
circuit optimization via multiple use of a specific two-qubit gate.
