Quantum Fourier Transform: Phasor Sum (2D)

This is the 2D companion to the 3D Quantum Fourier Transform ring simulator, and it deliberately shows a different part of the same mathematics. Where the 3D scene renders only the finished amplitude of each output basis state as a needle's height and arrow, this simulator draws the actual complex-plane vector sum that produces one output amplitude c_j = (1/√N) Σ_x a_x·e^{dir·2πi·jx/N}: every non-zero input term becomes one phasor, chained tip-to-tail, with the resultant arrow being the real, computed c_j. Scrub the output index or sweep through all of them to watch the chain re-angle itself while a spectrum panel fills in the full output distribution bar by bar. Three input modes go beyond the 3D sim's single basis state: a two-state superposition shows constructive and destructive interference directly in the chain, and a period-r "comb" input reproduces the exact post-measurement register Shor's algorithm feeds into its QFT step — watch the transform concentrate all output probability onto multiples of N/r and read the hidden period straight off the peak spacing.