The quantum NOT gate can be considered the quantum counterpart of the classical NOT gate. The classical NOT gate inverts the value of a bit:

If the input is 0, the output is 1.
If the input is 1, the output is 0.

In quantum computing, this operation is performed on quantum states (qubits), and the quantum NOT gate extends the fundamental function of the classical gate. Let us delve deeper into this concept below.

4.3.2.1. Definition of the Quantum NOT Gate

The quantum NOT gate operates on quantum states. In the vector space C2, qubits with the basis states 0⟩ and ∣1⟩are processed by this gate as follows:

Input: ∣x⟩, where x ∈ {0,1}
Output: ∣1 ⊕ x⟩

Here, the operation ⊕ represents XOR. The output is described as:
If the input is ∣0⟩, the output is ∣1⟩ (since 1⊕0=1).
If the input is ∣1⟩, the output is ∣0⟩ (since 1⊕1=0).

Mathematically, this operation is defined as:
notq ∣x⟩ = ∣1 ⊕ x⟩, which inverts the value of the qubit.

4.3.2.2. Properties of the Quantum NOT Gate

Like the classical NOT gate, the quantum NOT gate is a reversible operation. This means that applying the gate again yields the original input:
notq (notq ∣x⟩)= ∣x⟩

This exemplifies one of the fundamental properties of quantum computing: reversibility. Reversibility is one of the most crucial principles of quantum operations and enables the use of this gate in quantum mechanics.

4.3.2.3. Measurement and Outcomes of the Quantum NOT Gate

n quantum systems, the measurement process determines the quantum state, causing the system to collapse into a definite state after measurement. When the quantum NOT gate operates on the basis states ∣0⟩ and ∣1⟩, the measurement outcomes are as follows:

Input ∣0⟩: The quantum NOT gate transforms this state into ∣1⟩. If this new state (∣1⟩) is measured, the measurement result definitively collapses the system to ∣1⟩, and this outcome is observed.

Input ∣1⟩: The quantum NOT gate transforms this state into ∣0⟩. If this new state (∣0⟩) is measured, the measurement result definitively collapses the system to ∣0⟩, and this outcome is observed.

During quantum measurement, the system transitions from a quantum state to a classical state. That is, after the measurement, the system permanently transitions to the state resulting from the measurement:

Input ∣0⟩ and measurement outcome ∣1⟩: Initially, the system was in state ∣0⟩ ; the NOT gate transformed it to ∣1⟩, and upon measuring, the system collapses to the classical state ∣1⟩.

Input ∣1⟩∣1⟩ and measurement outcome ∣0⟩: Initially, the system was in state ∣1⟩; the NOT gate transformed it to ∣0⟩, and the measurement results in the system collapsing to the classical state ∣0⟩.

4.2.3.4. Superposition Situation

In quantum computing, qubits do not exist solely in the basic states ∣0⟩ and ∣1⟩. A qubit can also be in a superposition state:

a₀|0⟩ + a₁|1⟩

where a₀ and a₁ are complex numbers representing the probabilities of the qubit being in the respective basis states.

When the quantum NOT gate is applied to such superposition states, each component is inverted individually:
notq (a₀|0⟩ + a₁|1⟩) = a₀|1⟩ + a₁|0⟩

The way this gate acts on superposition states illustrates how quantum gates differ from and surpass classical gates in complexity and power. These properties of superposition enable quantum computers to be significantly more powerful than classical computers, allowing them to perform complex computations simultaneously.