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Quantum Fourier Transform

The quantum analogue of the Discrete Fourier Transform — central to Shor's algorithm and quantum phase estimation. Step through the circuit and watch how Hadamard and controlled-rotation gates transform computational basis states into superpositions with structured phase relationships.

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Circuit Diagram

What's Happening

State Vector — Amplitudes

Magnitude Phase Changed

Phase Wheels

Composite Phasor Diagram

State in Dirac Notation

Binary Fraction Decomposition — The Key Insight

Tensor Product Structure

QFT Unitary Matrix (FN) — highlighted column = input |x⟩

Frequency Spectrum — DFT Interpretation