Let $P(iharrj)$ be the matrix exchanging the $i$ th and $j$ th rows $($columns$)$ of vectors$/$matrices$.$ For example,

on a four$-$dimensional space,

$P(1$harr$2)=\left[\begin{array}{ccc}0& 1& ?\\ 1& 0& ?\\ ?& 1& ?\\ ?& 1& ?\end{array}\right].$

widehat$(P)(iharrj)$ is the corresponding operator.

$($a$)$ Find a quantum circuit for widehat$(P)(2$harr$3)$ on a two$-$qubit system $($four$-$dimensional vector space SoxS $)$

using CNOT gates only.

$($b$)$ Find a quantum circuit for widehat$(P)(2$harr$3)$ on a three$-$qubit system $(S^{ox3})$ using multi$-$control NOT gates

only.

$($c$)$ Find a quantum circuit for widehat$(P)(4$harr$7)$ on a three$-$qubit system $(S^{ox3})$ using multi$-$control NOT gates

only.