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The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.
This gate is the first example of a controlled quantum gate that you should study.
Figure 4. CNOT gate symbol. Source. The symbol follow the generic symbol convention for controlled quantum gates shown at Figure 3. "Generic controlled quantum gate symbol", but replacing the generic "U" with the Figure 2. "Quantum NOT gate symbol".
To understand why the gate is called a CNOT gate, you should think as follows.
First let's produce a generic quantum state vector where the control qubit is certain to be 0.
On the standard basis:
we see that this means that only and should be possible. Therefore, the state must be of the form:
where and are two complex numbers such that
If we operate the CNOT gate on that state, we obtain:
and so the input is unchanged as desired, because the control qubit is 0.
If however we take only states where the control qubit is for sure 1:
Therefore, in that case, what happened is that the probabilities of and were swapped from and to and respectively, which is exactly what the quantum NOT gate does.
So from this we understand more concretelly what "the gate only operates if the first qubit is set to one" means.
Now go and study the Bell state and understand intuitively how this gate is used to produce it.

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  1. Controlled quantum gate
  2. Multi-qubit gate
  3. Single-qubit gate
  4. Quantum logic gate
  5. Digital quantum computer
  6. Analog and digital quantum computers
  7. Model of quantum computing
  8. Quantum computer type
  9. Quantum computing hardware
  10. Quantum computing
  11. Quantum information
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