Toffoli Gate Matrices

Question

Here are the different toffolis (or maybe one of them is toffoli and the others are very similar to toffoli gates)

My question is:

1. we know the matrix of the number 1 Toffoli:

What are the matrices for other two toffoli gates?

Answer 1

The problem is much easier to think about in turns of permutations. You can see that the matrix you created simply swaps the last two elements of the statevector: $[a_{000}, a_{001}, a_{010}, a_{011}, a_{100}, a_{101}, a_{110}, a_{111}]$ becomes $[a_{000}, a_{001}, a_{010}, a_{011}, a_{100}, a_{101}, a_{111}, a_{110}]$

The middle gate above swaps $a_{010}$ and $a_{011}$. The right gate swaps $a_{100}$ and $a_{101}$. The matrix is the identity matrix, but the two $1$s corresponding to these two rows are moved to the opposite corners of the square they are corners of.

And yes, the two gates, together, make an odd parity check.

Answer 2

                             -----UPDATED------

Thanks to @Rajiv Krishnakumar, I solved the problem by this link Here are my toffolis:

The other two toffoli was already found yesterday:

Answer 3

I always find enlightening the block-matrix display approach, so that these three Toffoli operators can be viewed as

$$\begin{bmatrix}I& & & \\ &I& & \\ & &I& \\ & & &X\end{bmatrix}$$ $$\begin{bmatrix}I& & & \\ &X& & \\ & &I& \\ & & &I\end{bmatrix}$$ $$\begin{bmatrix}I& & & \\ &I& & \\ & &X& \\ & & &I\end{bmatrix}$$

where $I$ is the $2$x$2$ identity operator, $X$ is the NOT operator, and the white spaces are all zeroes for tidyness.