Question 3 (25 marks, learning outcome 02) = (a) Let X := |0)(1| + |1)(0|, Y := i |0)(1| + i |1)(0], and Z := |00|11| be the three Pauli matrices, and let I |0)(0| + |1)(1| be the identity matrix. Show that the matrix U := a I+i (bX+cY+dZ) is unitary for all real numbers a, b, c, d satisfying the condition a +62 +c + d = 1. (5 marks). (b) A qubit, initially in the state |0), undergoes the unitary gate U with a = b = 1/2 and c = d=0. What is the state after the gate has acted? (5 marks) (c) After the gate has acted, the qubit of point (b) is measured in the computational basis {|0), |1)}. What are the probabilities of the outcomes? (5 marks) (d) Alice and Bob play a game. Alice prepares a qubit in the state |0) and sends it to Bob. Bob promises to Alice that he will either apply the unitary gate U (with probability 1/2), or do nothing (with probability 1/2). Bob sends the qubit back to Alice, without telling her what he did. Alice's task is to guess whether he applied the gate U or not. To this purpose, Alice measures in the computational basis {|0), |1)} and makes a guess: if the outcome is 0, she guesses that Bob did nothing, and if the outcome is 1, she guesses that Bob applied the gate U. What is the total probability that Alice's guess is correct? (5 marks) (e) Consider the game of the previous point, with the only difference that now, instead of mea- suring in the computational basis {|0), |1)}, Alice measures in the Fourier basis {|+), |)}, with |) := (|0) 1))/2. What is the probability the Alice's guess is correct now? [Hint: fix a guessing strategy for Alice: for example, if she finds outcome +, she guesses that Bob did nothing, and if she finds outcome " she guesses that Bob applied the gate U. Try different guessing strategies and see if the answer depends on the strategy you choose] (5 marks)