Improve the usability of callback functions in ground state optimization

[20akshay00]

When playing around with different solvers and new algorithms to compute the ground state of a system, we typically want to track certain properties including but not limited to the energy and galerkin error over the iterations of the solver. Currently there is find_groundstate which takes in an algorithm, and the algorithm takes in an argument for a callback function where the state of the solver is exposed and can be altered. As I am working with FiniteMPS, I have only looked into DMRG and GradientGrassmann so far;

• GradientGrassmann expects finalize!(x, f, g, numiter) -> x, f, g
• DMRG expects finalize(iter, state, H, envs) -> state, envs

In order to keep track of some quantities during the optimization, something along these lines has to be done:

function custom_find_groundstate(args...)
  history = []
  function custom_finalizer(iter, state, H, envs) # specific to DMRG
      current_error = maximum(pos -> calc_galerkin(state, pos, envs), 1:length(state))
      current_energy = expectation_value(state, H, envs)
      custom_observable = expectation_value(state, ...)
      push!(history, (; current_error, current_energy, custom_observable, t=Base.time()))
      return state, envs
  end
  
  return find_groundstate(args...; finalize=custom_finalizer), history
end

Alternatively, if some kind of local state is to be preserved, something like this can also be done:

mutable struct IterationData
      energy
      error
      observable
      IterationData() = new([], [], [])
end

function (data::IterationData)(iter, state, H, envs) # specific to DMRG
   push!( data.error, maximum(pos -> calc_galerkin(state, pos, envs), 1:length(state)))
    push!(data.energy, expectation_value(state, H, envs))
    push!(data.observable, expectation_value(state, ...))
    return state, envs
end
  
custom_data = IterationData()
psi, envs, delta = find_groundstate(args...; finalize=custom_data), history

While this is functional, it is a bit clunky and also has to be modified per algorithm as they expect functions with different signatures. There may also be some minor inconsistencies as in #207. Following are possible improvements:

• Ideally there should be a uniform interface for the signature of finalize, although I am unsure if there is a clean way to do so for these two methods (and any other added later on) given that they operate with different quantities. For example, always having access to at least H (or E = <H>, but consistently either one of them) and psi within finalize makes sense to me.
• Custom data can be returned as the third return value from find_groundstate instead of just delta.
• A way to quit the optimization loop in find_groundstate using a custom convergence criteria instead of a hard-coded one which is also solver dependant :/.

My use case for this is really just to quantitatively judge the performance of Gradient methods vs. DMRG for my system in order to decide which one to use hereafter. I think this would be a fairly common use case for anyone working with atypical systems when DMRG is no longer sufficient, so having a convenient way to do this would be quite helpful.