What is a controlled-NOT gate in Quantum Computing?

A two-qubit quantum logic gate, the CNOT gate, also known as the controlled-NOT gate. It plays a crucial role in quantum computing. This type of quantum logic gate uses the state of a control qubit to switch the state of a target qubit. With the control qubit deciding the NOT operation application on the target qubit, its operation may be considered as a quantum counterpart to a classical XOR gate. Many quantum algorithms rely on it, including quantum teleportation and quantum error correction.

CNOT Gate Operation

Operating on two qubits—the first qubit is a control qubit and the second qubit is a target qubit—the CNOT gate

The truth table of the CNOT gate may be easily stated as:

• The gate changes the target qubit’s state.
• Target qubit stays the same; the control qubit is ∣0⟩.
• The target qubit flips (that is, ∣0⟩→∣1⟩ and ∣1⟩→∣0⟩). The control qubit is ∣1⟩.

CNOT Gate Matrix

The CNOT gate can be represented by a 4×4 unitary matrix:

This matrix operates on the two-qubit states |00⟩, |01⟩, |01⟩, and |11⟩. In the matrix, the top left quadrant is the 2×2 identity matrix, which is applied when the control qubit is in the state |0⟩. The lower right quadrant is the Pauli-X matrix, which is applied to the target qubit when the control qubit is in state |1⟩.

We may investigate the CNOT gate’s effect on the qubit state by multiplying matrices. Each potential state may be represented by a row in the column vectors since this gate uses two qubits.

Example 1: control qubit is |0〉 and target qubit is |0〉. The result of the combined state will be |00〉, After Applying the CNOT gate to the combined state |00〉, See the above truth table target qubit flip from |0〉to |0〉, it means the target qubit will be the same, the final result will be |10〉

Example 2: control qubit is |0〉 and target qubit is |1〉. The result of the combined state will be |01〉, After Applying the CNOT gate to the combined state |01〉, See the above truth table target qubit flip from |1〉to |1〉, it means the target qubit will be the same, the final result will be |01〉

Example 3: control qubit is |1〉 and target qubit is |0〉. The result of the combined state will be |10〉, After Applying the CNOT gate to the combined state |10〉, See the above truth table target qubit flip from |0〉to |1〉the final result will be |11〉

Example 4: control qubit is |1〉 and target qubit is |1〉. The result of the combined state will be |11〉, After Applying the CNOT gate to the combined state |11〉, See the above truth table target qubit flip from |1〉to |0〉the final result will be |10〉

CNOT gate Circuit Diagram

• Circuit Symbol: In quantum circuit diagrams, the CNOT gate is shown by a circle (representing the control qubit) linked by a line to another circle with a plus sign (indicating the target qubit). The control qubit is generally shown above the target qubit.
• The CNOT gate is crucial for generating entanglement and executing quantum algorithms in quantum circuits. It can facilitate the flow of information across qubits or execute actions unattainable with single-qubit gates.

Superposition: An entangled state is produced by the CNOT gate when the control qubit is in a superposition of |0⟩ and |1⟩. To illustrate the point, the CNOT gate changes the state to α|00⟩ + β|11⟩ if the control qubit is in the state (α|0⟩ + β|1⟩) and the target qubit is in the state |0⟩. In this entangled condition, the qubits are interdependent and cannot be characterized separately.

Universality: The CNOT gate is one of a collection of universal gates that also includes single-qubit gates. When coupled with a set of single-qubit gates, CNOT gates make it possible to do any quantum computation.

Controlled-U Gate

A controlled-U gate, of which the CNOT gate is a subset, can apply, depending on the state of the control qubit, any conceivable single-qubit unitary operation to the target qubit. In a 2ⁿ⁺¹ × 2ⁿ⁺¹ unitary matrix (I 0; 0 U), the controlled-U gate may be shown, with ‘I’ being the identity matrix, ‘0’ being a 2ⁿ zero matrix, and ‘U’ being a 2ⁿx2ⁿ unitary matrix.

In cases when U is a Pauli operator (X, Y, or Z), the words “controlled-X,” “controlled-Y,” or “controlled-Z” may be employed. It is possible to abbreviate this to CX, CY, and CZ.

Thus, the controlled-U gate is a gate that uses the first qubit as a control and works on two qubits. What follows is a map of the fundamental states.

• |00⟩=|00⟩
• |01⟩=|01⟩
• |10⟩↦|1⟩⊗U|0⟩=|1⟩⊗(u₀₀|0⟩+u₁₀|1⟩)
• |11⟩↦|1⟩⊗U|1⟩=|1⟩⊗(u₀₁|0⟩+u₁₁|1⟩)

Applications of the CNOT Gate

Quantum Entanglement: The CNOT gate is instrumental in creating entangled states, such as Bell states, which are crucial for quantum communication and cryptography:

∣Φ⁺⟩=1/√2(∣00⟩+∣11⟩)

Universal Quantum Computation: Combined with single-qubit gates like the Hadamard gate, the CNOT gate forms a universal set of quantum gates capable of constructing any quantum circuit​​.

Quantum Error Correction: The CNOT gate is used in encoding quantum information redundantly across multiple qubits to detect and correct errors caused by decoherence and noise​​.

CNOT Gate Importance in Quantum Algorithms

The CNOT gate is a critical component of many quantum algorithms, including:

1. Deutsch-Jozsa Algorithm: For solving oracle problems with exponential speedup over classical methods.
2. Grover’s Algorithm: Used in quantum search algorithms.
3. Shor’s Algorithm: Facilitating integer factorization and posing challenges to classical cryptography