Canonical construction of Logical Fourier Gate

Question

For physical $d$-dimensional qudits we can define $$X= \sum_{i=0}^{d-1} |i+1\rangle \langle i |$$ and $$Z = \sum_{i=0}^{d-1} \omega^i |i\rangle \langle i |,$$ with $\omega=e^{2\pi i/d}$. The Fourier gate $$F=\frac{1}{\sqrt{d}} \sum_{i,j=0}^{d-1} \omega^{ij} |i\rangle \langle j|$$ transforms one basis into the other, i.e. $F X F^\dagger = Z$.

Now consider a stabilizer error correction code. One can choose the logical $Z_L$ operator such that it commutes with all stabilizer generators, and one can choose $X_L$ such that it commutes with the stabilizer generators plus it fulfills $Z_L X_L = \omega X_L Z_L$. That should fix our logical qudit space. Now we can choose any operator that maps $X_L$ to $Z_L$ as our $F_L$, but is there any canonical way to construct it? I was hoping that there is some compact formula for $F_L$ in terms of $Z_L$ and $X_L$, but I couldn't find it.

To be clear, I am not asking for a decomposition of $F_L$ in terms of a circuit (even though that is interesting as well), but I am "only" looking for a canonical way to construct the unitary operator itself.

Answer 1

There is no circuit for $F$ in terms of $X$ and $Z$, because $\{X, Z\}$ is not a universal set of gates and $F\notin\langle X,Z\rangle$. The first assertion follows from the fact that the action of any product of $X$ and $Z$ on a state in the computational basis is in the computational basis. The second one follows from the observation that the action of $F$ on any computational basis state is a uniform superposition of computational basis states.

That said, we can express $F$ as a polynomial in $X$ and $Z$. First note that

$$ \frac1d\sum_{k=0}^{d-1}Z^k = \frac1d\sum_{k=0}^{d-1}\left(\sum_{l=0}^{d-1}\omega^l|l\rangle\langle l|\right)^k = \frac1d\sum_{l=0}^{d-1}\left(\sum_{k=0}^{d-1}\omega^{kl}\right)|l\rangle\langle l| = |0\rangle\langle 0|. $$

Therefore,

$$ \frac1d\sum_{k=0}^{d-1}X^iZ^kX^{-j} = X^i|0\rangle\langle 0|X^{-j} = |i\rangle\langle j| $$

which provides a convenient means of expressing any matrix in terms of $X$ and $Z$. Substituting into the formula for $F$, we get

$$ F = \frac{1}{\sqrt{d}}\sum_{i,j=0}^{d-1}\omega^{ij}|i\rangle\langle j| = \frac{1}{d\sqrt{d}}\sum_{i,j,k=0}^{d-1}\omega^{ij} X^iZ^kX^{-j}. $$

If desired we can also eliminate the powers of $\omega$ using $\omega = X^{-1}ZXZ^{-1}$.