quantum gate directory

SDK Support

SDK	Name
Qiskit	qiskit.circuit.library.QFT
PennyLane	pennylane.QFT
Cirq	cirq.qft
Q#	Std.Canon.ApplyQFT
PyQuil	—
Braket	—
BQSKit	—

The quantum Fourier transform maps the computational basis state $|j\rangle$ to an equal superposition with phases determined by $j$. Let $N = 2^n$ and $\omega = \mathrm{e}^{2\pi i / N}$, then

$$ \mathrm{QFT}_{N} |j\rangle = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \omega^{jk} |k\rangle $$

For two qubits ($N = 4$, $\omega = i$):

$$ \mathrm{QFT}_{4} = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -i \\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i \end{bmatrix} $$

In general the matrix has entries $[\mathrm{QFT}_N]_{jk} = \frac{\omega^{jk}}{\sqrt{N}}$.

Properties

• Unitary: $\mathrm{QFT}_{n}^\dagger = \mathrm{QFT}_{n}^{-1}$.
• The inverse QFT (used in phase estimation readout) has conjugated phases $\omega^{-jk}$.
• Achieves an exponential speedup over the classical FFT for in-place computation on a quantum state; however, the amplitudes cannot be read out directly without measurement.