Exercise 2.1: Prove in several different ways that HH = I.
Solution: First we use matrix algebraic multiplication
Next we utilize the superposition principle which claims that performing a linear operator on a superposition is equivalent to computing the outcomes with each individual computational basis state as an input and than adding the results together. Since HH|0〉 = |0〉 and HH|1〉 = |1〉 therefore two consecutive Hadamard gates act as an identity transform on their arbitrary superpositions.
Finally we remember that Hadamard gates are Hermitian (H† = H) and unitary (H† = H−1) thus HH = HH† = HH−1 = I.
Exercise 2.2: Prove that HXH = Z, HYH = −Y and HZH = X.
Exercise 2.3: Perform the analysis of the generalized interferometer using the superposition principle.
Solution: We start from |0〉 according to Fig. 2.7. The first Hadamard gate produces
The phase shifter introduces delays independently along the two paths
Finally we apply the second Hadamard ...