The inverse quantum Fourier transform (IQFT) is the inverse Fourier-basis transform on an n-qubit state. Its amplitudes are computed as inner products with Fourier basis states, using rotations with negative exponents; operationally, it can turn phase progression in a geometric sequence state into a distribution of amplitude magnitudes. section 7.3.2 section 8.0
A gate implementation applies Hadamard gates and controlled phase rotations with negative angles, then optionally swaps qubits. It mirrors the QFT circuit except for rotation signs. The gate count grows quadratically with the number of qubits, while equivalent classical state-vector processing traverses exponentially many amplitudes. section 7.4 section 7.4
Figure 7.22 Circuit diagram of the IQFT on a four-qubit system
Inspect the four-qubit IQFT circuit to see the Hadamard, negative controlled-phase, and swap pattern. section 7.4
Recovering an encoded frequency
For the Fourier basis state F13 on four qubits, the IQFT recovers computational basis 1101. Processing starts at the rightmost qubit: a Hadamard and controlled phase rotations remove already-encoded phase contributions, revealing one binary digit per step; a final swap restores qubit order. section 7.4.1
Frequency encoding uses the IQFT in two important cases:
- An integer frequency such as v = 4 in a three-qubit geometric sequence becomes outcome 4 with amplitude magnitude 1 and all other magnitudes zero. section 8.2.1
- A non-integer frequency such as v = 4.7 produces magnitudes concentrated near neighboring integer outcomes, with the closest outcome largest. section 8.2.1
- Quantum phase estimation applies the IQFT to its estimation register, then uses measurement outcomes to estimate an encoded phase frequency. section 9.3
- Function encodings apply controlled phase rotations and then the IQFT to a value register, associating input selections with integer values or weights. section 10.0 section 10.1.2
