# Quantum Fourier Transform without SWAPs

The Quantum Fourier Transform from Nielsen and Chuang chapter 5 is pictured here: In the textbook the author refers to "swap gates at the end of the circuit which reverse the order of the qubits".

My questions are:

1. Is it possible to transform the circuit shown in some way to avoid the need for any SWAP gates while still using little-endian conventions. Naively, I might think I could "flip the circuit upside down" so that the first operation is H(n), then R2 on qubit n controlled by qubit (n-1), and so on...

2. The Wikipedia page on QFTs makes no reference to reordering or SWAP gates - does this imply a different bitstring convention between the sources, or an error in one of the sources?

• You could express the swap with controlled-rotations and Hadamards ... – Norbert Schuch Jun 10 at 21:23
• Do you know how that compares in gate depth assuming the circuit compiles each SWAP to three CNOTs? – forky40 Jun 10 at 21:37
• take a look this video youtube.com/watch?v=uuBgK44JrnA&t=2s – Aman Jul 22 at 11:43

With this notation, the action of the quantum Fourier transform can be expressed in a compact manner:$$QFT|x_1x_2\dots x_n⟩=\frac{1}{\sqrt{N}}(|0⟩+e^{2\pi i[0.x_n]}|1⟩)\otimes(|0⟩+e^{2\pi i[0.x_{n-1}x_n]}|1⟩)\otimes\dots\otimes(|0⟩+e^{2\pi i[0.x_1x_2\dots x_n]}|1⟩)$$