Quantum Fourier Transform circuit

[jimmycricket]

I'm having a very hard time understanding how the QFT can be realized using just the Hadamard and controlled rotation gates. Furthermore, I cannot see why swap gates are used to reverse the order of the qubits. I'm embarrassed that don't have much by way of any attempt to show here since I am so stumped by this but I can set up a scenario. Say we wish to perform the transformation on a 3 qubit system in the superposition state (omitting normalization factors)
[itex]\sum\limits_{x=0}^7|{x}\rangle= |{000}\rangle+|{001}\rangle+|{010}\rangle+...+|{111\rangle}=|{1}\rangle+|{2}\rangle+...+|{7}\rangle[/itex]

 

[eljose79]

Hello there,

I understand your confusion about using only Hadamard and controlled rotation gates to realize quantum Fourier transform (QFT) and why swap gates are used to reverse the order of qubits. Let me explain it in more detail.

Firstly, let's understand the concept of superposition and entanglement. In a quantum system, qubits can exist in multiple states at the same time, known as superposition. This allows for a greater range of possibilities and allows for more complex calculations to be performed. Entanglement, on the other hand, is a phenomenon where qubits become correlated with each other, even when separated by great distances. This means that the state of one qubit can affect the state of the other qubit, no matter how far apart they are.

Now, let's look at the QFT algorithm. It is a way to transform the state of a quantum system from one basis to another. In the case of a 3 qubit system, the basis states are |0⟩,|1⟩,|2⟩,...,|7⟩. The QFT algorithm transforms these basis states into the state |{0}\rangle+|{1}\rangle+...+|{7}\rangle, which is known as the Fourier basis. This transformation is essential for many quantum algorithms, such as Shor's algorithm for factoring large numbers.

To perform this transformation, we need to use Hadamard and controlled rotation gates. Hadamard gate is used to create superposition, while controlled rotation gates are used to entangle qubits. These two gates, when used together, can create any desired transformation. In the case of QFT, these gates are used to create the superposition of all basis states and entangle them in a specific way to produce the Fourier basis.

Now, why do we need to use swap gates to reverse the order of qubits? This is because in the QFT algorithm, the order of the qubits is important. The first qubit represents the most significant bit (MSB), while the last qubit represents the least significant bit (LSB). In order to perform calculations on these qubits, we need to reverse the order so that the MSB is at the end and the LSB is at the beginning. This is where the swap gates come in. They simply swap the positions of qubits, allowing us to perform calculations on the correct qubits.

I hope this explanation helps