The quantum Fourier transform (QFT) is the quantum analogue of the discrete Fourier transform (DFT): it applies a Fourier transform to the amplitudes of a quantum state, producing another quantum state. It switches between amplitude encoding and phase encoding, making hidden periodicity and other patterns easier to identify. section 5.3.4
Use QFT with these boundaries in mind:
- Apply it as a building block for phase estimation, especially when learning phases with one or two bits of precision. section 5.3.4
- Do not treat it as a faster classical DFT: state preparation and result extraction can make QFT less efficient than the fast Fourier transform. section 5.3.4
- For one qubit, recognize QFT as the Hadamard gate acting on a computational-basis state. section 5.3.4
One-qubit case
For a basis state |x⟩, the one-qubit QFT produces an equal superposition whose amplitudes carry the Fourier-transformed information; operationally, this is the Hadamard transform. section 5.3.4
