In the previous section, we discussed single-bit or single-qubit gates. Now, we will talk about 2-bit or 2-qubit states. Let’s assume we have two different gates, which we will refer to as Gate 1 and Gate 2. Now, let’s begin this section by discussing the properties of these gates.
4.4.1. CNOT and XOR Gate
Gate 1: Let this gate be an irreversible gate. In other words, this gate does not allow a return from an output state to the input state. This implies a loss of information or situations where the input cannot be precisely reconstructed.
Gate 2: Let this gate be a reversible gate. This means that it is possible to return to the original input information from the resulting output of this gate. Reversibility is one of the fundamental properties of quantum gates, as they can perform operations without losing information.
Now, if we consider having CNOT and XOR gates, we can define Gate 1 and Gate 2 as follows.
Gate 1: It can be an XOR Gate. The XOR gate, in classical computation, produces an output based on the input of two bits and is irreversible. This is because the same output can come from different input combinations. Additionally, the XOR gate takes a 2-qubit input.
Below is the truth table of the XOR Gate:
| Input (xy) | Output (x⊕y) |
|---|---|
| (0,0) | 0 |
| (0,1) | 1 |
| (1,0) | 1 |
| (1,1) | 0 |
As seen in the table above, the XOR gate produces the same result for the inputs (0,1) and (1,0). It is not possible to determine which input combination was used by looking at the output state. Since each output value corresponds to more than one input combination, the function is not reversible.
As shown in the table above, the irreversible XOR gate only produces the result of ( x ⊕ y ) and loses the original inputs.
Gate 2: It can be a CNOT Gate. The CNOT gate is a reversible gate in quantum computation. It operates on two qubits, but unlike the XOR gate, it has a control bit and a target qubit. In this case, we can determine the input state by looking at the output.
The CNOT gate, which is a reversible gate, produces both ( x ) and ( x ⊕ y ) as outputs, allowing the original inputs to be recovered.