Controlled-U Gate

Definition

The Controlled-U gate is a two-qubit gate that applies a unitary gate $U$ on the target qubit only when the control qubit is in the state $|1\rangle$. If the control qubit is in the state $|0\rangle$, the target qubit remains unchanged. In simple words, the Controlled-U gate is a generalized version of the CNOT gate because the CNOT gate is just a special case where the unitary gate $U$ is the Pauli-X gate.

PS: The word "controlled" means there is one qubit which decides whether the operation should happen or not. The word "U" means any valid single-qubit unitary operation can be placed there.

Effect on qubit

The Controlled-U gate does not change the control qubit. It only checks the value of the control qubit.

1. If the control qubit is $|0\rangle$, nothing happens to the target qubit.
2. If the control qubit is $|1\rangle$, the gate $U$ is applied to the target qubit.

This is why the Controlled-U gate is extremely useful in quantum algorithms. It allows us to make one quantum operation depend on the state of another qubit. That conditional behavior is one of the main reasons multi-qubit circuits become powerful.

Types

The Controlled-U gate can have many types because the gate $U$ can be any unitary single-qubit gate. Some common special cases are:

• Controlled-X gate: This is the CNOT gate.
• Controlled-Z gate: This applies the Pauli-Z gate conditionally.
• Controlled-H gate: This applies the Hadamard gate conditionally.
• Controlled-S gate: This applies the S gate conditionally.

Representation

Matrix representation

Suppose the single-qubit unitary gate is

$$U = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \tag{1}$$

Then the matrix representation of the Controlled-U gate is:

$$CU = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{bmatrix} \tag{2}$$

By carefully observing this matrix, you can see that the first two basis states $|00\rangle$ and $|01\rangle$ remain unchanged. The last two basis states $|10\rangle$ and $|11\rangle$ are the states where the control qubit is $|1\rangle$, so the gate $U$ acts there.

Circuit representation

The Controlled-U gate is represented as:

---o---
   |
---U---

Where o represents the control qubit and U represents the unitary operation applied to the target qubit.

Example

Let us take a simple example where the unitary gate is the Hadamard gate. That means we are applying a Controlled-H gate.

Suppose we have two qubits initially in the state $|10\rangle$. The first qubit is the control qubit and the second qubit is the target qubit.

$$|q_0\rangle = |1\rangle \tag{3}$$ $$|q_1\rangle = |0\rangle \tag{4}$$

So the combined state is:

$$|q_0q_1\rangle = |10\rangle \tag{5}$$

The Hadamard gate is:

$$H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \tag{6}$$

Therefore, the Controlled-H matrix is:

$$CH = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \tag{7}$$

To apply the Controlled-H gate to the state $|10\rangle$, we write the input state vector as:

$$|10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \tag{8}$$

Now perform the matrix multiplication:

$$CH|10\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \tag{9}$$

Simplifying:

$$CH|10\rangle = \begin{bmatrix} 0 \\ 0 \\ \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} \tag{10}$$

This means:

$$CH|10\rangle = \frac{1}{\sqrt{2}}(|10\rangle + |11\rangle) \tag{11}$$

Hence, because the control qubit was $|1\rangle$, the Hadamard gate got applied to the target qubit.

If instead the input state had been $|00\rangle$, then the output would still be $|00\rangle$ because the control qubit would be $|0\rangle$.

Properties

The properties of the Controlled-U gate are:

• The Controlled-U gate is unitary if $U$ is unitary.
• The control qubit is not changed by the gate.
• The Controlled-U gate becomes a CNOT gate when $U = X$.
• The Controlled-U gate becomes a Controlled-Z gate when $U = Z$.
• The Controlled-U gate is important in phase estimation, quantum Fourier transform based circuits, and many variational quantum circuits.

Conjugate Transpose

The conjugate transpose of the Controlled-U gate is:

$$CU^\dagger = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & a^* & c^* \\ 0 & 0 & b^* & d^* \end{bmatrix} \tag{12}$$

Here $a^*, b^*, c^*, d^*$ represent the complex conjugates of the corresponding entries.

Inverse

The inverse of the Controlled-U gate is:

$$(CU)^{-1} = C(U^{-1}) \tag{13}$$

This means the inverse of a Controlled-U gate is simply the controlled version of the inverse of $U$.

Dagger

The dagger of the Controlled-U gate is:

$$(CU)^\dagger = C(U^\dagger) \tag{14}$$

So if the original gate on the target is replaced by its dagger, we get the dagger of the whole Controlled-U gate.