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Suppose there is a superposition state $|{{\Phi }^{+}}\rangle =\sum\limits_{i=0}^{15}{U({{\theta }_{i}})|i\rangle }$, I want to get $\langle i|U{{({{\theta }_{i}})}^{\dagger }}U({{\theta }_{m}})|m\rangle ,i\ne m,i,m\in [0,15]$. The swap test or Hadamard test is aimed at between two superposition states. Is it suitable for local inner product of superposition states? How can I explain my needs in terms of quantum circuits?

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    $\begingroup$ If I tell you a state $|\Phi^+\rangle$, can you give a unique decomposition in terms of $U(\theta_i)$? $\endgroup$
    – DaftWullie
    Sep 12, 2022 at 6:40

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