 Dyadic quantum Fourier transform circuit

In most-significant-bit-first order, let $x=x_1\cdots x_n$ and $0.x_j\cdots x_n={\sum }_{l=j}^n{x}_l{2}^{-(l-j+1)}$. On a computational-basis state, process wire $j$ with a Hadamard gate, then controlled phase gates $R_{l-j+1}$ controlled by unprocessed wire $l$ for each $l>j$. Wire $j$ becomes $(|0⟩+e^{2\pi i0.x_j\cdots x_n}|1⟩)/\sqrt{2}$. Reversing the final wire order yields the positive-exponent quantum Fourier transform. There are $n$ Hadamard gates and $n(n-1)/2$ controlled phase gates. Physical reversal costs $\lfloor n/2\rfloor$ swap operators, each expressible as three controlled-NOT gates, themselves implemented with Hadamard gates and a Controlled-Z gate. Thus the entire construction uses $O(n^2)$ gates from the specified family.