The Quantum Fourier Transform (QFT) matrix looks very similar to the DFT matrix except with a normalization factor of . The matrix for the Quantum Fourier transform on qubits is

where and .

In the following codercises, we will implement the QFT for one- and two-qubit systems.

Implement the circuit that performs the single-qubit QFT operation, i.e., for .

Hint.

For , , and .

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Learning Objectives:

  • Derive the Quantum Fourier Transform (QFT) for one and two qubits.
  • Describe the properties of the QFT.
  • Express the QFT for one and two qubits in terms of gates.