Is there a smart way to implement $e^{i\theta\,\Phi\,\rm{QFT} \, \Phi \, \rm{QFT}^\dagger}$, where both $\Phi \propto\sum_j2^jZ_j$ and $\rm{QFT}$ act on the same set of registers? Even an approximate implementation valid for small $\theta$ would be helpful.
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It shouldn't be guaranteed that you can do this. For example lets look at the 1 qubit case $QFT = H, \Phi=Z$. We now have $\Phi QFT \Phi QFT^\dagger = ZHZH = ZX$ which is antihermition. The overall exponent is now Hermitian and the resulting matrix wont be unitary.
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