Please refer to the following questions

2. 4 marks Find a 1-qubit state vector |+y) such that Y|ty) = Ity). Find a 1-qubit state vector (-y) such that Y|-y) = -|-x). 3. 3 marks Prove that for [Boo) = $10)10) + 4(1)|1), and for any unitary one qubit gate G we have: GO I | Boo) = 18 G* | Boo) where G is the transpose of G. 4. 3 marks Give a protocol that wins the following game with probability . I An adversary does one of the following (with no restriction on the probability with which she chooses option 1 or 2): 1) Gives you either |0) or (1), with equal probability. OR 2) Gives you 410) + 4211). You must guess whether the adversary performed 1) or 2). Note that you only get to play the game once. Hint: First come up with a protocol that has 'one-sided' error. That is, for one outcome of the protocol, the guess is always correct, but for the other outcome, it may be wrong sometimes. Then post-process so that the error probability is equal in both cases