Speed up dynamics computation

[cerisola]

This bug report is to keep track of ideas/tests to work towards speeding up the computation of the dynamics.

[cerisola]

As a first test, I would suggest to create the initial state state0, the Liouvillian super operator superop, and then instead of solving the differential equation, run:

u0 = vec(state0)
du = similar(u0)
@time dstate(du, u0, nothing, 1.0)

and post here the results.

Similarly, also run:

u0 = vec(state0)
du = similar(u0)
@code_warntype dstate(du, u0, nothing, 1.0)

and post the results.

Based on the outcomes we can try to see if there is some unnecessary allocations/type-instabilities that are slowing down the differential equation solving.