RZZ from CNOT and RZ

Question

The following should represent an RZZ gate (source: https://pennylane.ai/qml/demos/tutorial_qaoa_maxcut.html)

How do the CNOT and an RZ compute mathematically to the RZZ?

$$ R_Z(\theta) = \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 \\ 0 & e^{i\frac{\theta}{2}} \end{pmatrix} $$

$$ CNOT = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix} $$

The above circuit, if I understand, should result in the following:

$$ R_{ZZ}(\theta) = \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 & 0 & 0 \\ 0 & e^{i\frac{\theta}{2}} & 0 & 0 \\ 0 & 0 & e^{i\frac{\theta}{2}} & 0 \\ 0 & 0 & 0& e^{-i\frac{\theta}{2}} \end{pmatrix} $$

Answer 1

The $R_Z(\theta)$ operator is in reality the $I \otimes R_Z(\theta)$ operator which makes it $$ \begin{align} I \otimes R_Z(\theta) &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \otimes \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 \\ 0 & e^{i\frac{\theta}{2}} \end{pmatrix} \\ &= \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 & 0 & 0 \\ 0 & e^{i\frac{\theta}{2}} & 0 & 0 \\ 0 & 0 & e^{-i\frac{\theta}{2}} & 0 \\ 0 & 0 & 0 & e^{i\frac{\theta}{2}}\end{pmatrix} \end{align} $$ This is because if you have to include the identity operator when looking at the overall 2-qubit system. Now you can perform the matrix multiplication

$$ \begin{align} CNOT\left(I \otimes R_Z(\theta)\right)CNOT &= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix} \begin{pmatrix} e^{-i\frac{\theta}{2}} & 0 & 0 & 0 \\ 0 & e^{i\frac{\theta}{2}} & 0 & 0 \\ 0 & 0 & e^{-i\frac{\theta}{2}} & 0 \\ 0 & 0 & 0 & e^{i\frac{\theta}{2}}\end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix} \end{align} $$ which will give you the $R_{ZZ}(\theta)$ matrix you have written (up to a global phase).

Answer 2

It helps to write out the matrix representation of the RXX gate you post in the beginning. First, we can easily see that

$$(I - Z_1Z_2)/2 = \begin{pmatrix} 0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0 \end{pmatrix}$$ So looking at $R_{ZZ}(\theta) = e^{-i\theta(I-Z_1Z_2)/2}$, we can again easily see that $$ R_{ZZ}(\theta)=\begin{pmatrix} 1&0&0&0\\ 0&e^{-i\theta}&0&0\\ 0&0&e^{-i\theta}&0\\ 0&0&0&1 \end{pmatrix}. $$ Now, you correctly conclude the resulting matrix from the circuit above, getting $$ R_{ZZ}(\theta)=\begin{pmatrix} e^{-i\frac{\theta}{2}}&0&0&0\\ 0&e^{i\frac{\theta}{2}}&0&0\\ 0&0&e^{i\frac{\theta}{2}}&0\\ 0&0&0&e^{-i\frac{\theta}{2}} \end{pmatrix}, $$ which you'll notice we can rewrite as $$ R_{ZZ}(\theta)=e^{-i\frac{\theta}{2}}\begin{pmatrix} 1&0&0&0\\ 0&e^{i\theta}&0&0\\ 0&0&e^{i\theta}&0\\ 0&0&0&1 \end{pmatrix}, $$ Giving us the same matrix as above up to some global phase differences, which we can generally ignore.