docs: improve split-operator documentation

[Lewiscowles1986]

I've added a high level description which I think better classifies and summarises this approach to solving a problem.

I've used the term meta algorithm, because this can be applied to a wide number of problems, with different, contextual and situational concrete algorithms.

[leios]

Wait, I might be confused here. So my understanding was that:

1. The split operator method almost always refers to what's presented in the text as is. That is, a specific method that splits up the operators in a hamiltonian to solve them in their specific spaces. The split-step Fourier method is the same algorithm under a different name
2. The split-step method (without the Fourier distinction) is as you say, a general class of algorithms that can be used in a variety of different situations.

[Lewiscowles1986]

Maybe I need to further improve this, and I've over-genericised the text.

If I'm honest, I was a little... What? while reading this, so I was trying to simplify from a process of collating a few sources.

• Accelerating the Fourier split operator method via graphics processing units
• OPERATOR SEPARATION OF VARIABLES FOR ADIABATIC PROBLEMS IN QUANTUM AND WAVE MECHANICS
• Noisy intermediate-scale quantum simulation of the one-dimensional wave equation
• Quantum Lattice Boltzmann is a quantum walk
• Hamiltonian formulation and symplectic split-operator schemes for time-dependent
  density-functional-theory equations of electron dynamics in molecules

I'd welcome feedback if I've made the text harder to understand, or misunderstood any of the materials, because I am not a quantumn theoretical computer scientist, or theoretical physicist

[leios]

so I think I am starting to see the issue here.

1. Though "Split operator method" is usually the split-step fourier method, there are many people who use the term to refer to operator splitting, in general. Once the splitting is done, there is no need to really use the fourier transform for the solution.
2. This means that the title of this chapter should be "Split-step fourier method" instead of "split-operator method".

[leios]

I think I am understanding where you are coming from this statement. I see some of the references still had operator splitting, but did not specifically use the Split-step fourier method. Even though in practice the SSFM is the most common approach used for the split operator method, I think it should be made more explicit in the text that there is a difference between these approaches.

[Lewiscowles1986]

So I have a question here. Should I make the change in my branch to call this, "split-step fourier method", or have I really just taken us both down a rabbit hole which is unhelpful?

[leios]

Can I come back to this? I think what might happen is that I will create a new PR that is co-authored with you with the necessary changes. This discussion is a good one, I just have to rethink about how to package it.

It'll take like 10 minutes, but I might not have that time available until later this week.

[Lewiscowles1986]

its not urgent to me. This was sort of me thinking I was helping, but maybe I'm not

[leios]

For the record, the conversation is definitely helpful! Thanks