In the 1995 paper 'Almost any Quantum Logic Gate is Universal' by Seth Lloyd, the author imagines a setup where we can apply two Hamiltonians $A$ and $B$ to a quantum system. By repeatedly and alternately applying these Hamiltonians, a unitary of the following form can be performed on the system: \begin{align} U = ...e^{i B t_4 }e^{i A t_3 }e^{i B t_2 }e^{i B t_1 } \tag{1} \end{align} where $t_n$ are the times for which each Hamiltonian is applied. The author then claims that any unitary of the form $e^{i L t}$ can be generated this way, where $L$ is a member of the algebra generated from $A$ and $B$ through commutation.
I did not understand the proof of this statement as my knowledge of Group Theory is very rudimentary, so I tried to prove it for the simple case where $A=\sigma_z$ and $B=\sigma_x$ and the algebra in question is $SU(2)$ (where $\sigma_{x,y,z}$ are the Pauli matrices). If I have understood the result correctly, I believe that this should mean that we can generate any $2 \times2 $ unitary by repeated applications of the Hamiltonians $\sigma_z$ and $\sigma_x$ since $[\sigma_x, \sigma_z ] = 2 i \sigma_y$ and , together with the identity, $\{\sigma_x, \sigma_y, \sigma_z\}$ form a basis for the $2\times2 $ Hermitian matrices, which can be used to generate the $2\times 2$ unitary matrices. This is what I am trying to prove and cannot do it.
There are two proofs that I am interested in. The first is the 'existence' proof, which shows that any $2\times2$ unitary $U$ can be generated in the following way: \begin{align} U = ...e^{i \sigma_z t_4 }e^{i \sigma_x t_3 }e^{i \sigma_z t_2 }e^{i \sigma_x t_1 } . \tag{2} \end{align}The second question is that of a construction proof, ie. given $U$, how can we find $t_1, t_2$ etc. such that $(2)$ is satisfied?
Any help on either of these questions would be appreciated.
