# Is there a way to find the inner product between some ground states in a superposition?

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?

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