The Inverse QFT

The inverse QFT (written $\mathrm{QFT}^\dagger$ or $\mathrm{IQFT}$ ) undoes the quantum Fourier transform. It is an essential building block of quantum phase estimation and Shor's algorithm: after the QFT encodes a phase into register amplitudes, the inverse QFT reads that phase back out as a measurable bit string.

Inverting a unitary circuit

Every quantum gate is unitary, so its inverse is its conjugate transpose. For a product of unitaries the reversal rule is

(U_1 U_2 \cdots U_k)^\dagger = U_k^\dagger \cdots U_2^\dagger U_1^\dagger.

To invert a circuit, reverse the gate order and replace each gate with its adjoint. Two useful special cases:

The Hadamard is self-inverse: $H^\dagger = H$ .
The controlled-phase gate conjugates as $CP(\lambda)^\dagger = CP(-\lambda)$ , because negating the angle complex-conjugates the phase $e^{i\lambda} \to e^{-i\lambda}$ .

The 3-qubit inverse QFT

Recall the 3-qubit forward QFT (without final bit-reversal SWAPs), with qubit 0 as the most significant bit:

| Step | Gate | |------|------| | 1 | $H(q_0)$ | | 2 | $CP(\pi/2),\; \text{ctrl}=q_0,\; \text{tgt}=q_1$ | | 3 | $CP(\pi/4),\; \text{ctrl}=q_0,\; \text{tgt}=q_2$ | | 4 | $H(q_1)$ | | 5 | $CP(\pi/2),\; \text{ctrl}=q_1,\; \text{tgt}=q_2$ | | 6 | $H(q_2)$ |

Applying the reversal rule gives the inverse QFT:

| Step | Gate | |------|------| | 1 | $H(q_2)$ | | 2 | $CP(-\pi/2),\; \text{ctrl}=q_1,\; \text{tgt}=q_2$ | | 3 | $H(q_1)$ | | 4 | $CP(-\pi/4),\; \text{ctrl}=q_0,\; \text{tgt}=q_2$ | | 5 | $CP(-\pi/2),\; \text{ctrl}=q_0,\; \text{tgt}=q_1$ | | 6 | $H(q_0)$ |

The structure is a mirror image of the forward circuit: the coarsest qubit ( $q_2$ ) is processed first, and the phase rotations go in the negative direction.

Why it matters

In quantum phase estimation, a register is left in a state whose amplitudes encode a phase $\varphi$ as $e^{2\pi i k \varphi / 2^n}$ for each basis state $|k\rangle$ . Applying $\mathrm{QFT}^\dagger$ transforms that "Fourier-encoded" state back to a computational-basis state close to $|2^n \varphi\rangle$ , making $\varphi$ directly readable by measurement. Without the inverse QFT, the phase information would be locked in the amplitudes and inaccessible.

The gate count of $\mathrm{QFT}^\dagger$ is identical to that of the QFT: $n$ Hadamards and $n(n-1)/2$ controlled-phase gates, totalling $O(n^2)$ operations for an $n$ -qubit register.

Try it

Implement the 3-qubit $\mathrm{QFT}^\dagger$ by reversing the forward QFT's gate sequence and negating all phase angles. The grader checks the full unitary matrix — leaving the circuit empty or incorrect will fail.

Run your code to see the quantum state.