Quantum Circuit Confusion On Time Evolution

[thatboi]

Hi all,
When working in the Heisenberg picture, we can represent implementing time evolution on an operator via a Hamiltonian H through a quantum circuit type picture like the following:

where time is on the vertical axis and increases going up and the block represents the unitary gate ##e^{-iHt}##. However, I am struggling to picture how this would look on a circuit if instead, we wanted to conjugate some operator ##O## via the unitaries ##e^{-iH_{1}t},e^{-iH_{2}t}## where ##[H_{1},H_{2}] \neq 0##. That is, ##O(t) = e^{-iH_{2}t}e^{-iH_{1}t}Oe^{iH_{1}t}e^{iH_{2}t}##. Vertically stacking the gates on top of each other doesn't seem to make much sense to me since it would then seem to imply that we have elapsed a time ##2t## through this time evolution.
Any thoughts?

 

[Demystifier]

Is it really important how much time was spent? If not, then vertically stacking gates is OK. If yes, then you need to find an operator ##H## such that
$$e^{iH_1t}e^{iH_2t}=e^{iHt}$$
For that purpose, you need to use some version of Baker-Campbell-Hausdorff formula.