Expanding two rotation operators

Question

I have the following operators acting on two qubits, denoted as $(1)$ and $(2)$:

$$T_1=\displaystyle\exp\left(-i\frac{\pi}{4}Z\otimes Z\right)\cdot R_z^{(1)}\left(\frac{\pi}{2}\right)\cdot R_z^{(2)}\left(\frac{\pi}{2}\right),\quad T_2=R_y^{(2)}\left(-\frac{\pi}{2}\right)\cdot C_Z\cdot R_y^{(2)}\left(\frac{\pi}{2}\right).$$

I should prove that up to global phases $T_1=C_Z$ and $T_2=C_X$ (the controlled $Z$ and $X$ gates).

For convenience, I omit the labels for the qubits, which should be clear by the ordering. Using the usual expansion for rotations, I ended up finding

$$T_1=\frac{1}{2\sqrt 2}(\mathbb I\otimes\mathbb{I}-iZ\otimes Z)(\mathbb I\otimes \mathbb I-iZ\otimes\mathbb I)(\mathbb I\otimes \mathbb I-i\mathbb I \otimes Z)=\frac{1+i}{\sqrt 2}\begin{pmatrix} -Z \\ & \mathbb I\end{pmatrix},\\ T_2=\frac{1}{2}(\mathbb I\otimes \mathbb I+i\mathbb I\otimes Y)C_Z(\mathbb I\otimes \mathbb I-i\mathbb I\otimes Y)=\begin{pmatrix} \mathbb I & \\ & X\end{pmatrix}.$$ So, I've checked the calculations a few times and it looks like there is something wrong with the first operator. Here's the catch: there was a mistake in the original $T_2$ I was given that I had to change in order to find the correct sequence to obtain a $C_X$, and there might be a similar problem with $T_1$ (unless I am of course making some mistake somewhere). Does anyone see what the issue may be?

(As a reference, $T_1$ comes from the interaction shift in NMR computing, while $T_2$ from Wineland's experiment with trapped ions).

Answer 1

I agree with your calculation for $T_1$ and $T_2$ starting from your expansion of the operations (the second set of equations). Since there's a couple of possible different conventions for the definition of, for example $R_z(\theta)$, I cannot be certain that the expansion itself is correct. Indeed, it requires relatively minor modification to make things work. For example, if you replace both of the $\pi/2$ rotation angles in the statement for $T_1$ with $-\pi/2$ rotation angles, it all works happily.

It may assist you to know the way that I do this calculation. For $T_1$, you've got 3 matrix exponentials, all of which are using diagonal operators. This means they all commute and hence can be grouped together as a single exponential. $$ T_1=e^{-i\pi Z_1Z_2/4}e^{i\pi Z_1/4}e^{i\pi Z_2/4}=e^{i\pi(Z_1+Z_2-Z_1Z_2)/4} $$ So I can just concentrate on the diagonal elements of these, $$ Z_1+Z_2-Z_1Z_2=\text{diag}(1,1,1,-3) $$ I can add (or subtract) identity matrices as this only contributes a global phase $$ Z_1+Z_2-Z_1Z_2-I=\text{diag}(0,0,0,-4). $$ So when I multiply by $\pi/4$, I'll get a $\pi$ for the final element which, exponentiated, gives the -1 I need.