QFT resource screen

The QFT maps computational basis states to a Fourier basis: $\mathrm{QFT}|j\rangle=\frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1}e^{2\pi i jk/2^n}|k\rangle$ for an $n$-qubit register. The standard circuit applies a Hadamard on the top qubit, then controlled phase rotations $R_k=\mathrm{diag}(1,e^{2\pi i/2^k})$ from every later qubit into it, repeats for each qubit in turn, and optionally reverses qubit order with a final SWAP layer depending on bit-order convention — the layer this record's metadata keeps explicit. The circuit uses $n$ Hadamards and $\binom{n}{2}$ controlled-phase gates, giving $O(n^2)$ total gates versus the classical FFT's $O(N\log N)=O(2^n\cdot n)$ arithmetic operations on a full amplitude vector — an exponential gate-count saving, but only over quantum amplitudes that must then be sampled, not a free classical output vector. This record's check is a parser/compile-compatibility comparison across the export path, not a numerical circuit-correctness proof.