**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*.

*Solution:*

**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 ...

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