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The QFT on the group $\mathbb{Z}_N$ is given by \begin{equation} QFT\,|k\rangle =\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi i\,jk/N}|j\rangle\,. \end{equation}

The usual circuit implements the QFT with $N=2^n$ on $n$ qubits. Does anyone know how to realize this transformation when $N$ is not a power of 2? I have found an algorithm that approximates a QFT with arbitrary $N$, but I would like to find out if some exact circuit is known. I think that it should at least in some cases, since the problem 5.1 of Nielsen and Chuang's book is to find it when $N=p$ is a prime number.

I'm interested in particular on the $\mathbb{Z}_3$ transform, which corresponds to the matrix \begin{equation} \frac{1}{\sqrt{3}}\,\begin{pmatrix} 1&1&1 \\ 1&e^{2\pi i/3}&e^{-2\pi i/3} \\ 1&e^{-2\pi i/3}&e^{2\pi i/3} \end{pmatrix}\,. \end{equation}

Of course it acts on a 3d space, so to realize it using qubits we should extend it to a unitary operator on at least a $2^2$ dimensional Hilbert space. One possibility could be

\begin{equation} \begin{pmatrix} \frac{1}{\sqrt3 } \begin{pmatrix} 1&1&1 \\ 1&e^{2\pi i/3}&e^{-2\pi i/3} \\ 1&e^{-2\pi i/3}&e^{2\pi i/3} \end{pmatrix} & \\ & 1 \end{pmatrix}\,, \end{equation} but then how can we realize it exactly? When $N=2^n$ we can get the $\frac{1}{\sqrt{2^n}}$ prefactor with the n Hadamard gates that prepare the uniform superpositions, but now that $\sqrt{3}$ factor is really bothering me.

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Kitaev gave an algorithm to approximate Quantum Fourier Transform over an arbitrary cyclic group.

Later, Mosca and Zalka showed how his construction can be made exact.

You can find a nice description for Kitaev algorithm in Dave Bacon lecture notes for example.

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