 Solution

The Hamiltonian operator annihilates every vector orthogonal to both of its defining states:Every positive power also annihilates that vector. The exponential series therefore givesIt is a stationary zero-energy state under this unitary time evolution. The displayed evolution uses $\hslash =1$, as in the question. For $N=1$ the two independent defining vectors already span the whole qubit space, so no nonzero vector satisfies both orthogonality conditions; for larger $N$ their orthogonal complement has dimension $2^N-2$.