# Are $\mathrm{cRz}$ with angle $\pi$ and $\mathrm{cZ}$ equal on the IBM Quantum Experience?

I was trying to implement the 2-qubit Grover algorithm without the auxiliary bit. Before finding the gate symbol for the controlled $$\mathrm{Z}$$, I worked with the controlled $$\mathrm{Rz}$$. The gate glossary states: " $$\mathrm{Z}$$ is equivalent to $$\mathrm{Rz}$$ for the angle $$\pi$$". So I thought the controlled $$\mathrm{Rz}$$ with angle $$\pi$$ should be equal with the controlled $$\mathrm{Z}$$ gate. However, I get very different results in the browser state-vector representation.

So my question is, should they be equivalent?

Here are my circuits:

I realized that $$\mathrm{Rz}(\lambda)$$ is implemented in following way on IBM Q:

gate crz(lambda) a,b
{
u1(lambda/2) b;
cx a,b;
u1(-lambda/2) b;
cx a,b;
}


Setting $$\lambda =\pi$$, a matrix describing construction above is following: $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -i & 0 \\ 0 & 0 & 0 & i \\ \end{pmatrix}$$

This is not problem when the gate is not controlled as $$i$$ is a global phase, however, it matters for controlled gates.

I also checked that these values are really returned for computation basis states in state vector visualization on IBM Q.

Conclusion is that controlled $$\mathrm{Rz}(\pi)$$ is not equivalent to controlled $$\mathrm{Z}$$ on IBM Q.

Note that application of single qubit $$\mathrm{Rz}(\pi)$$ returns same results as single qubit $$\mathrm{Z}$$.

Solution:

To make controlled $$\mathrm{Rz}(\pi)$$ behave as expected, you have to put controlled global phase gate before $$\mathrm{Rz}(\pi)$$. You can do that by application single-qubit gate $$\mathrm{U1}(\pi/2)$$ on controlling qubit of $$\mathrm{Rz}(\pi)$$, i.e. $$q_0$$ in your case.

• You say this is a bug, but I wouldn't call it so. The definition follows very naturally from the single-qubit definitions. The bug is the expectation that they should be the same. They shouldn't because of the difference between global and relative phase. Feb 28 '20 at 12:10
• @DaftWullie: I would expect that controlled version of single qubit gate $\mathrm{U}$ is described by matrix $\begin{pmatrix} I & O \\ O & U\end{pmatrix}$ as e.g. in case of controlled $\mathrm{X}$, $\mathrm{Z}$ or $\mathrm{H}$. Feb 28 '20 at 12:45
• It is, i's just the $U$ that you're talking about is equivalent to $Z$ up to a global phase, but it is not $Z$. Feb 28 '20 at 12:53
• No, because $R_z$ is not $Z$. $R_z=-iZ$. Feb 28 '20 at 13:29
• This, I can agree with. Feb 28 '20 at 13:37