CNOT Gates

Let’s start with the $\text{CNOT}$ gate, introduced for classical logic gates in the Logic Gates and Truth Tables and Gates as Matrices lessons. The quantum circuit schematic is shown in the image below that shows what we might expect for a quantum $\text{CNOT}$ gate based on what we know about a classical-computing $\text{CNOT}$ gate. We will see that the correspondence is correct if the states $|A⟩$ and $|B⟩$ are the computational basis states $|0⟩$ and $|1⟩$, but the results are more interesting (and more useful) for more general quantum states. Recall that the upper-qubit line is called the control qubit and the lower-qubit line is the target or controlled qubit.

First, we need to translate from classical bits $0$ and $1$ to quantum state vectors $|0⟩$ and $|1⟩$. The logic table for the classical $\text{CNOT}$ gate is given below.