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There is a known circuit construction https://arxiv.org/abs/quant-ph/0208130 Also check this post https://algassert.com/post/1710 It uses phase estimation ideas to derive the resulting circuit


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Existence (as mentioned on Wikipedia, for example), is easy, although not necessarily unique. Any unitary can be written as $$ U=\sum_ie^{i\theta_i}P_i $$ where $P_i$ are projectors such that $\sum_iP_i=1$, and $\theta_i$ is in the range 0 to $2\pi$. Then we have that $$ \sqrt{U}=\sum_ie^{i\theta_i/2}P_i. $$ Giving a circuit construction of the gate is an ...


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