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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.

Write both the input and output integer labels in most-significant-bit-first order, $x={\sum }_{j=1}^n{x}_j{2}^{n-j}$ and $y={\sum }_{j=1}^n{y}_j{2}^{n-j}$. Factoring the Fourier phase over the bits of $y$ gives the product decomposition of a Fourier phase state:Here $0.x_l\cdots x_n$ denotes the binary fraction ${\sum }_{r=l}^n{x}_r/2^{r-l+1}$; the integer part of $x/2^j$ does not affect the phase.

Build a dyadic quantum Fourier transform circuit as follows. Process wires $j=1,\dots ,n$ in that order. First apply a Hadamard gate to wire $j$. Then, for each $l=j+1,\dots ,n$, apply the controlled phase gate $R_{l-j+1}$ with control wire $l$ and target wire $j$. On a computational-basis state, every unprocessed control is still the bit $x_l$, so wire $j$ becomesOnce a wire has been processed it is not touched by later stages. Reversing the order of the output wires gives exactly the Fourier product above. Agreement on every computational-basis state proves agreement as linear operators on all superpositions, even though the intermediate operation on a general input can create entanglement.

There are $n$ Hadamard gates and $n(n-1)/2$ controlled phase gates. Output reversal may be treated as wire relabelling. If it must instead be implemented physically while using only the requested gate family, use $\lfloor n/2\rfloor$ swap operators. Each swap is three controlled-NOT gates, andThus the swaps use only Hadamard gates and Controlled-Z gates, which are controlled $R_1$ gates. An exact physical network haselementary gates from the stated family. This implements the positive-exponent Fourier transform with the correct output order in quadratic size. Omitting the output reversal without relabelling the wires would produce the reversed-order transform instead.

For the perturbation bound, the operator norm in the source is equivalently $\Vert A\Vert ={\sup }_{\Vert \psi \Vert =1}\Vert A\psi \Vert$. It satisfies the triangle inequality, is unchanged by left or right multiplication by a unitary operator, and a unitary operator has norm one. The exact telescoping identity isIt follows by replacing one factor at a time; alternatively all intermediate products cancel in the sum. Taking the operator norm and using unitary invariance gives the quantum circuit gate-error telescoping boundThe original PDF assumes strict $\Vert U_j-V_j\Vert <ϵ$, so its strict conclusion is justified. If the hypothesis were weakened to a non-strict inequality, the conclusion would likewise be $\le mϵ$. The TeX aid misreads this hypothesis; the PDF resolves it.

Apply the non-strict version to each approximate unitary gate in the Fourier network. Extending a gate by the identity on other wires leaves its operator norm error unchanged. Including the physical swap decompositions, all gates are of the required type and can obey the stated accuracy. HenceWith exact output relabelling rather than physical swaps the numerator is just $n(n+1)/2$. The proof retains gate phases and makes no assumption of cancellation between coherent errors.

For every normalized input the output-state norm error is at most this same bound. If $M$ is any positive operator satisfying $0\le M\le I$, the change in its measurement in quantum mechanics outcome probability is at most $2\delta$, where $\delta =\Vert U_n-{\mathrm{QFT}}_{2^n}\Vert$: expand the difference of the two expectations into two terms, each bounded by $\delta$ using the Cauchy-Schwarz inequality. Thus inverse-polynomial gate accuracy gives an inverse-polynomial total error, rather than requiring exponential precision in the number of qubits. For the phase gates this corresponds to specifying angles with $O(\log n)$ bits of precision, since $|e^{i\theta }-e^{i{\theta }^{\prime }}|\le |\theta -{\theta }^{\prime }|$. This is a robustness statement for unitary gate approximations; it does not by itself correct quantum decoherence, correlated stochastic faults, or limitations of hardware connectivity.