The commutation matrix $K^{(r,m)}$ is defined by
$ K^{(r,m)} = \sum_{i=1}^r \sum_{j=1}^m (\boldsymbol{e}_{r,i} \boldsymbol{e}_{m,j}^T) \otimes (\boldsymbol{e}_{m,j} \boldsymbol{e}_{r,i}^T) $
It also has the convenient property that $K^{(r,m)}(A\otimes B)K^{(n,q)}=B\otimes A$ where $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{r\times q}$.
By playing around, I found the following:
$K^{(2,2)}=SWAP(0,1) \quad$ algassert 1
$K^{(2,4)}=SWAP(0,1)\,SWAP(1,2) \quad$ algassert 2
$K^{(4,2)}=SWAP(1,2)\,SWAP(0,1) \quad$ algassert 3
It seems that $K^{(r,m)}$ is implemented with purely $SWAP$ gates. This brings me to my question, is there a general circuit implementation for $K^{(r,m)}$?
