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Unexpected Relative Phase while making the Toffoli Gate

Quantum Computing Asked by Zyad Yasser on February 5, 2021

I was trying to build the Toffoli gate using the following diagram (found in the qiskit textbook):
qiskit book toffoli gate diagram

So, I set V := Rx(pi/2) (as shown in the following diagram) (Note: that I am switching now to the little-endian format)

However, the unitary matrix equivalent to the previous was

$$
begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -i\
0 & 0 & 0 & 0 & 0 & 0 & -i & 0\
end{pmatrix}
$$

That will lead to a relative phase of $(-i)$ in the states $lvert 110 rangle$ and $lvert 111 rangle$

What is the reason for that?

One Answer

$R_x(pi/2)$ isn't a square root of the X gate. If you square it, you get $-iX$ instead of $X$. That's where the phase is coming from.

Answered by Craig Gidney on February 5, 2021

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