Quantum Fourier Transform (QFT) | Amplitude Frequency Domain
Quick Technical Answer:
The Quantum Fourier Transform (QFT) performs a discrete Fourier transform on quantum amplitudes, mapping basis state |j⟩ to an equal superposition of phase-encoded basis states in O(n²) quantum gates compared to classical FFT O(n 2ⁿ).
Formula / Unitary:
|j\rangle \mapsto \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n-1} e^{2\pi i j k / 2^n} |k\rangle
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Frequently Asked Questions
Can we directly read out the Fourier transform amplitudes from QFT?
No. The transformed frequencies are encoded in the quantum amplitudes, not classical memory. We must use interference (e.g. in QPE or Shor's algorithm) to sample the dominant frequency with high probability.