SWAP Gate

Definition

The SWAP gate is a two-qubit gate that swaps the states of two qubits. The SWAP gate is a fundamental gate in quantum computing and is used to exchange the states of two qubits.

Effect on qubit

The SWAP gate swaps the states of two qubits. If the first qubit is in the state $|a\rangle$ and the second qubit is in the state $|b\rangle$, the SWAP gate changes the state of the first qubit to $|b\rangle$ and the state of the second qubit to $|a\rangle$. Swapping the states of two qubits does not mean exchanging the qubits themselves. The SWAP gate is a conditional gate that acts on two qubits. It just swaps the states and not the properties of the qubits.

PS: Unlike the CNOT gate, the SWAP gate does not require a qubit to be in a state of $|1\rangle$ to perform the swap operation. The SWAP gate swaps the states of the qubits regardless of their initial states.

Types

The SWAP gate has only one type.

Representation

The matrix representation of the SWAP gate is:

Matrix representation

$$SWAP = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{1}$$

PS: By carefully observing the matrix, you can see that the SWAP gate is a symmetric gate. And its somewhat similar to the CNOT gate.

Symbolic Representation

$$SWAP(|\psi_1\rangle \otimes |\psi_2\rangle) = |\psi_2\rangle \otimes |\psi_1\rangle \tag{2}$$

Circuit representation

The SWAP gate is represented as

───X───
   │
───X───

Where the first qubit is represented by the first X and the second qubit is represented by the second X.

Example

Let's take an example to demonstrate the SWAP gate.

• SWAP Gate

Suppose we have two qubits initially in the state $|01\rangle$, represented as:

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

The SWAP gate is represented by the following matrix:

$$SWAP = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{5}$$

To apply the SWAP gate to the qubits $|q_0\rangle = |0\rangle$ and $|q_1\rangle = |1\rangle$, we perform a matrix multiplication of the SWAP gate matrix with the state vector representing $|01\rangle$.

$$SWAP|q_0q_1\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \tag{6}$$

Performing the matrix multiplication:

$$SWAP|q_0q_1\rangle = \begin{bmatrix} 1*0 + 0*1 + 0*0 + 0*0 \\ 0*0 + 0*1 + 1*0 + 0*0 \\ 0*0 + 1*1 + 0*0 + 0*0 \\ 0*0 + 0*1 + 0*0 + 1*0 \end{bmatrix} \tag{7}$$

Simplifying:

$$SWAP|q_0q_1\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \tag{8}$$

Thus, the SWAP gate changes the state of the qubits $|01\rangle$ to $|10\rangle$.

Hence, we can say that:

$$SWAP(|01\rangle) = |10\rangle \tag{9}$$ $$SWAP(|10\rangle) = |01\rangle \tag{10}$$ $$SWAP(|11\rangle) = |11\rangle \tag{11}$$ $$SWAP(|00\rangle) = |00\rangle \tag{12}$$

We can conclude that the SWAP gate swaps the states of the qubits. The SWAP gate is a symmetric gate that swaps the states of two qubits without changing the properties of the qubits.

Properties

The SWAP gate has the following properties:

• The SWAP gate is a symmetric gate.
• The SWAP gate swaps the states of two qubits.
• The SWAP gate does not change the properties of the qubits.

Conjugate Transpose

The conjugate transpose of the SWAP gate is the same as the SWAP gate itself. The conjugate transpose of the SWAP gate is:

$$SWAP^{\dagger} = SWAP \tag{13}$$

Inverse

The inverse of the SWAP gate is the same as the SWAP gate itself. The inverse of the SWAP gate is:

$$SWAP^2 = I \tag{14}$$

Where $I$ is the identity matrix.

Dagger

The dagger of the SWAP gate is the same as the SWAP gate itself. The dagger of the SWAP gate is:

$$SWAP^{\dagger} = SWAP \tag{15}$$