Exponential of the product of unitary and hamiltonian matrices

Question

If I have a unitary diagonal matrix $U$ and a Hamiltonian $H$ (let's say it is time independent), can I do that manipulation? $$e^{i UHU^{\dagger} t}=U e^{i H t}U^{\dagger}$$

Answer 1

Yes, you can, because $$ e^{iUHU^\dagger t} = \sum_k \frac{(it)^k}{k!}(UHU^\dagger)^k = U\left(\sum_k \frac{(it)^k}{k!}H^k\right)U^\dagger = Ue^{iHt}U^\dagger $$ where we used the fact that $(UHU^\dagger)^k = UHU^\dagger \cdots UHU^\dagger = UH^kU^\dagger$ thanks to the property of unitarity $U^\dagger U =1$.