Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 58 / 4 / a / i / Solution

For $n\ge 2$, the Pauli X gate and Pauli Z gate act on different qubits in $X_0{Z}_1$, so they commute. Their product is a Hermitian matrix and squares to the identity operator. Its eigenvalues have modulus one; equivalently it is a unitary operator. Therefore its spectral norm isThe $n\ge 2$ qualification is necessary because the qubit labelled one does not exist for $n=1$.