 Solution

On an $N$-dimensional register, the quantum Fourier transform with positive-exponent convention isIt is the unitary version of the discrete Fourier transform on amplitudes. The inner product of columns $x,x^{\prime }$ is $N^{-1}{\sum }_y{e}^{2\pi i(x^{\prime }-x)y/N}$, equal to one when $x=x^{\prime }$ and zero otherwise by a finite geometric series, so the transformation is unitary.

For the qubit implementation take $N=2^n$, write $x=x_1\cdots x_n$ in most-significant-bit-first order, and put $0.x_j\cdots x_n={\sum }_{l=j}^n{x}_l{2}^{-(l-j+1)}$. Expanding $y={\sum }_j{y}_j{2}^{n-j}$ separates the Fourier phase as a product. The resulting dyadic quantum Fourier transform circuit identity isIndeed the phase on output bit $y_j$ is $e^{2\pi ix{y}_j/2^j}$, depending on the last $j$ input bits; multiplying the factors recovers $e^{2\pi ixy/2^n}$ and the normalization $2^{-n/2}$.

Implement the product in reverse wire order as follows. Process input wire $j$, starting at $j=1$. A Hadamard gate maps $|x_j⟩$ to $(|0⟩+e^{2\pi i{x}_j/2}|1⟩)/\sqrt{2}$. For each still-unprocessed wire $l>j$, apply a controlled phase gate with control $l$, target $j$, and target phaseOn a computational-basis input this multiplies the target's one-component by $e^{2\pi i{x}_l/2^{l-j+1}}$, so the processed wire becomes exactly $(|0⟩+e^{2\pi i0.x_j\cdots x_n}|1⟩)/\sqrt{2}$. Continue through all wires and reverse their order using SWAP gates. This equals the boxed transform on every basis vector and therefore on arbitrary superpositions by linearity; it does not require learning the input bits by measurement.

There are $\boxed{n\text{ Hadamards},n(n-1)/2\text{ controlled phase gates},\lfloor n/2\rfloor \text{ swaps}}$. A swap is three controlled-NOT gates, since the alternating control directions send $(a,b)$ successively to $(a,a\oplus b)$, then $(b,a\oplus b)$, and finally $(b,a)$. Thus the exact transform uses $O(n^2)$ one- and two-qubit gates from this phase-gate family, polynomial in the register size $n={\log }_{2}N$. This is a circuit-complexity statement, not a claim that all $N$ transformed amplitudes can be read out efficiently. If a fixed finite gate alphabet is required, the phase rotations must instead be synthesized to the desired accuracy; the exact count treats them as the available simple rotation gates.