Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 15D / b / i / Solution

For two nonorthogonal input states $|\alpha ⟩,|\beta ⟩$, preservation of inner products by $G$ givesHence the two output-program states agree up to phase. Any two qubit states can be joined through a state nonorthogonal to both, so after absorbing phases
$|{\tilde{P}}_U⟩$ is independent of the input.