Quantum Fourier Transform | mysimulator.uk

Qubits n 3 (N=8)

Input State |k⟩

Basis index k 0

Transform Direction

Output State

Σ_j c_j|j⟩ — spike at |0⟩, c₀=1

Dimension N

ΣP (unitarity)

1.000

Δφ / step

0.0°

How it works

The Quantum Fourier Transform is the discrete Fourier transform implemented as a unitary operation on qubit amplitudes. Acting on a computational basis state it produces:

QFT|k⟩ = (1/√N) Σ_{j=0}^{N-1} e^{ 2πi jk/N } |j⟩
IQFT|k⟩ = (1/√N) Σ_{j=0}^{N-1} e^{-2πi jk/N } |j⟩

|c_j| = 1/√N for every j   (equal magnitude)
arg(c_j) = 2π jk/N (mod 2π)   (linearly winding phase)

Each needle in the ring is one basis state |j⟩. Stem height = |c_j| (probability amplitude magnitude); the horizontal arrow atop it points at the complex phase arg(c_j), colour-coded the same way (hue = phase angle). Before the transform, all probability mass sits in a single spike at |k⟩ with phase 0. After Apply Transform, the amplitude spreads to every output state with identical magnitude 1/√N, but the phase advances by a fixed step Δφ = 2πk/N between adjacent states — exactly the "clock hand" winding that makes the QFT a frequency detector.

Qubits n — register size; N = 2ⁿ basis states (rendered on the ring).
Basis index k — which computational basis state |k⟩ is fed into the transform.
Forward / Inverse — sign of the exponent; the inverse QFT winds phase the opposite direction and is the step used to read out a phase-estimation register.
ΣP — sum of |c_j|² over all states, shown live as a sanity check that the transform stays unitary (always 1).

Real-world relevance: the QFT is the engine inside Shor's factoring algorithm and quantum phase estimation — after a controlled-unitary encodes a periodic phase into the register, the QFT converts that period into a measurable peak in the output distribution, giving quantum computers their speed-up for period-finding problems.

Stem height = |c_j| Arrow = phase arg(c_j)

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