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

enter image description here

My question is:

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

enter image description here

What are the matrices for other two toffoli gates?

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  • $\begingroup$ I would recommend to write down a map how three-qubit basis states (000, 001, 010, 011, 100, 101, 110, 111) are changed with the gates. Then convert the resulting basis states to vector representation and put them into a matrix (as columns) in the same order as the input states in the bracket above. $\endgroup$ Nov 28, 2022 at 7:10
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    $\begingroup$ An answer to a very similar question which will show you how to answer your own question :) quantumcomputing.stackexchange.com/questions/28315/… $\endgroup$ Nov 28, 2022 at 12:49
  • $\begingroup$ very helpful, thanks :) $\endgroup$
    – quest
    Nov 28, 2022 at 23:15

3 Answers 3

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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.

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  • $\begingroup$ From the three examples you have, you should be able to understand the pattern of which two states swap on a CNOT. I really think you should figure this out for yourself, to show you understood my answer. $\endgroup$ Nov 28, 2022 at 17:46
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                             -----UPDATED------

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

enter image description here The other two toffoli was already found yesterday: enter image description here

enter image description here

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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.

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