Quantum Fourier Transform

The Quantum Fourier Transform (QFT) is the subroutine that gives quantum computers their exponential speed-up for period-finding problems — it is the final step of Shor's factoring algorithm and the core of quantum phase estimation. This simulator renders the amplitude of every basis state |j⟩ in an n-qubit register as a 3D needle arranged in a ring: stem height encodes the amplitude's magnitude |c_j| and a horizontal arrow (colour-coded by hue) encodes its complex phase. Start from a single computational basis state |k⟩ — all probability mass in one spike — apply the forward or inverse QFT, and watch the amplitude spread into the transform's signature equal-magnitude, linearly-winding-phase pattern, with a live unitarity check (ΣP) and the exact phase step Δφ = 2πk/N shown alongside.