Phase estimation on $|0\rangle^{\otimes 2n}|\psi_j\rangle$: output $|\phi_j\rangle|\psi_j\rangle$ with $p_k \ge 4/\pi^2$ at the optimal $k$

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Phase estimation on $|0\rangle^{\otimes 2n}|\psi_j\rangle$: output $|\phi_j\rangle|\psi_j\rangle$ with $p_k \ge 4/\pi^2$ at the optimal $k$ · Prove2Me