Trouble proving approximation for quantum simulation using Zassenhaus formula

Question

I would like to prove the following $e^{i(A+B)\Delta t} = e^{\frac{iA\Delta t}{2}}e^{iB\Delta t}e^{\frac{iA\Delta t}{2}} + \mathcal{O}(\Delta t ^3)$. I have tried to prove it using Zassenhaus formula but i didn't succeed. Can anyone please help with this proof? Thank you

Answer 1

Simplify notation by $\Delta t \equiv t$.

The Zassenhaus formula is wild overkill. Just use the well-known leading CBH expansion, after multiplying both sides of your target expression by $e^{ {-iA t/2}} $, $$e^{ {-iA t/2}} e^{it(A+B) } - e^{itB }e^{ iA t/2} \\ =e^{ it(B+A /2) +t^2[A,B]/4 + \mathcal{O}( t ^3)} - e^{ it(B+A /2) +t^2[A,B]/4 + \mathcal{O}( t ^3)} = \mathcal{O}( t ^3).$$ The right hand side will be unaffected by multiplication by $e^{ iA t/2}$.