11. Gates: CNOT
Controlled-NOT (CNOT)
You have mastered the single-qubit rotations. Now you need the tool that links them.
The Controlled-NOT (CNOT) is the single most important multi-qubit gate, as it is the only non-local operation required to achieve universality in quantum computing.
1. The Matrix and the Operation
The CNOT gate acts on two qubits: a Control qubit and a Target qubit.
- The Rule: The state of the Target qubit is flipped (NOT) if and only if the Control qubit is $|1\rangle$. Otherwise, nothing happens.
The Matrix (Control Qubit 0, Target Qubit 1):
The CNOT is a $4 \times 4$ unitary matrix:
$$CNOT = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$The $2 \times 2$ blocks along the diagonal show the conditional action:
- The upper-left $2 \times 2$ block is the Identity ($I$) matrix, acting when the Control is $|0\rangle$.
- The lower-right $2 \times 2$ block is the Pauli-X ($X$) matrix, acting when the Control is $|1\rangle$.
This block structure is often written as:
$$CNOT = |0\rangle \langle 0| \otimes I + |1\rangle \langle 1| \otimes X$$2. Physical Significance: Creating Entanglement
The CNOT gate is a reversible gate that acts on a specific basis. Crucially, when the Control qubit is in a superposition (e.g., $|+\rangle$), the CNOT links the two qubits' states, transforming a product state into an entangled state.
This ability to couple the computational basis with the NOT operation is what creates the non-local correlations of entanglement.
Your Task: Entanglement by CNOT
As demonstrated when building the Bell states, the CNOT gate is the essential tool for linking the two qubits.
Start with the product state $|\psi\rangle$ where the Control qubit ($Q_0$) is in superposition and the Target qubit ($Q_1$) is $|0\rangle$:
$$|\psi\rangle = |+\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)$$Apply the $CNOT$ gate (Control $Q_0$, Target $Q_1$) to $|\psi\rangle$.
- How does $CNOT$ transform the $|00\rangle$ component?
- How does $CNOT$ transform the $|10\rangle$ component?
- What is the resulting state vector in Dirac notation? (This should be one of the Bell states you previously derived.)