Can a CNOT Gate Alone Create a Toffoli Gate?

[singedang2]

Homework Statement

I'm trying to show that controlled-not gate can not be used to make toffolit gate(controlled-controlled not gate)

Homework Equations

The Attempt at a Solution

i'm thinking that controlled not only has +(addition modulo 2) of the inputs, it can't make X and Y for input X,Y
hence cannot make function XY+Z

but I'm not sure if this is right, and I don't know how else to go about this problem.

thanks

 

[Jhero]

for any help!

You are on the right track with your thinking. The controlled-not gate, also known as the CNOT gate, is a two-qubit gate that performs an XOR operation on the control qubit and the target qubit. It flips the target qubit if and only if the control qubit is in the state |1⟩. This gate is commonly used in quantum computing to create entanglement and perform logical operations.

However, the CNOT gate alone cannot be used to create a Toffoli gate, also known as the controlled-controlled-not gate. The Toffoli gate is a three-qubit gate that performs an XOR operation on the first two qubits and then applies the result to the third qubit. This gate is commonly used in quantum computing for reversible computing and implementing classical logic gates.

To understand why the CNOT gate cannot be used to create a Toffoli gate, let's consider the truth table for each gate. The truth table for the CNOT gate is:

|Control | Target | Output |
|--------|--------|--------|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |

The truth table for the Toffoli gate is:

|Control 1 | Control 2 | Target | Output |
|----------|-----------|--------|--------|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |

As you can see, the CNOT gate does not have