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I am reading Grover's paper "Fixed-Point Quantum Search," (arXiv version with a different name) which improves on his earlier quantum research algorithm. However, I'm having difficulty in understanding some argument. How can we derive the first display on pp.4 from the last equation on pp.3 of the arXiv version. Many thanks!

2Welcome to the quantum stack exchange, could you please link the paper in the question ? – BrockenDuck – 2020-12-07T21:20:18.107

$U_{ts} = \langle t |U|s\rangle$ is the transition amplitude for the state $|s\rangle$ to be transformed to the state $|t\rangle$ under the action of the unitary $U$. Although it is poor notation, $\left\Vert U_{ts} \right\Vert_{}^{2}$ is simply the transition probability, namely, $U_{ts}^{2}$. Please be more specific about which derivation from Page 2 you don't understand and what you'd like to be clarified further; on the same note, please provide some context. – keisuke.akira – 2020-12-09T13:49:55.867