Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 1 / b / ii / Solution

The Pauli Y gate satisfies $Y^{†}=Y$ and $Y^2=I$. With $c=\cos (\pi /6)=\sqrt{3}/2$ and $s=\sin (\pi /6)=1/2$, the given matrix is Hermitian, andTherefore $M_2$ is not a unitary matrix. Its eigenvalues are $c+s$ and $c-s$, neither of which has modulus one. An expression $\cos \theta I+i\sin \theta Y$ would behave differently, but the printed matrix has no such factor of $i$.