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# 2. Qubit Mathematical Models. A qubit, ly), is in a state as given (differently in some cases) in the sub-questions at time to. The qubit evolves in time according to the following three time-dependent Hamiltonian in the following diagram at the times indicated. Please note that this diagram is NOT a "standard” quantum logic circuit as Hamiltonian are given and not the unitary transformation matrices.

b) (15 points for 5383/10 points for 7383) Assume that [y(to)) is initialized as given.Determine the value of ly) at time t2, ly(t2)). Show all work and clearly explain each step of your approach to find the result. \left|\Psi\left(t_{0}\right)\right\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}} e^{t \frac{\pi}{2}}|1\rangle c) (10 points for 5383/10 points for 7383) Determine the overall transfer matrix, U, that describes the time evolution of [y(t3))=U]y(to)). Clearly state your assumptions and-show all steps in your derivation of U. d) (7383 STUDENTS ONLY: 10 points for 7383) Assume that ly(t3)) evolves to state|y(t3))= |1). In this case, what was the initial state, ly(to))? Show all work and clearly-explain each step of your approach to find the result.

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