Help Understanding a Quantum Circuit Identity

[CMJ96]

Hello I have the following quantum circuit identity for converting a controlled U gate (4x4 matrix) into a series of CNOT gates and single qubit gates
$$ U= AXA^{\dagger}X$$
where A is a unitary matrix.
Here is a picture of the mentioned identity.

Can someone help me understand conceptually what is going on here? How do you actually define A?

 

[DrClaude]

Can someone help me understand conceptually what is going on here?

The idea is to use a CNOT gate and single qubit operations instead of trying to implement the controlled U. Single qubit operations are simpler to do in real experiments, so that the focus is to implement a CNOT gate and try to use it as much as possible. As you see, the price to pay is that you have to perform two CNOTs and two single-qubit operations instead of a single operation.

How do you actually define A?

Depends on what you want U to achieve. The idea is to find the A that allows you to end up in the same state as with the controlled U gate.

 

[CMJ96]

So when A is introduced, would it be another 4x4 unitary matrix (assuming the control U gate is a 4x4 matrix)?

 

[DrClaude]

So when A is introduced, would it be another 4x4 unitary matrix (assuming the control U gate is a 4x4 matrix)?

From $U= AXA^{\dagger}X$, you see that $U$ and $A$ have the same dimension. You can look at the action of $U$ and $A$ on the lower qubit only, in which case they are 2x2 matrices, but if you want the full controlled gate, which has to be a two-qubit operator, then you have to write them as a 4x4 matrices.

 

[CMJ96]

Hi I've gone away and had a think about this, and now I feel I understand what is happening pretty well, however I'm still struggling with applying it.
$$
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \frac{\sqrt{3}}{2} & \frac{-1}{2} \\
0 & 0 & \frac{1}{2} & \frac{\sqrt{3}}{2}
\end{bmatrix}
$$
Is it appropriate to be using the aforementioned circuit identity to write this controlled U as single qubit gates and CNOT's? I have been trying for a while now to figure out what unitary matrix fits into the equation for A and can't quite get it

 

[DrClaude]

Hi I've gone away and had a think about this, and now I feel I understand what is happening pretty well, however I'm still struggling with applying it.
$$
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \frac{\sqrt{3}}{2} & \frac{-1}{2} \\
0 & 0 & \frac{1}{2} & \frac{\sqrt{3}}{2}
\end{bmatrix}
$$

What basis are you using for the matrix representation? What matrix is that supposed to be? What is the U you are trying to apply to the second qubit?