Isolate indexing logic for noisy experiments

[pedrorrivero]

What is the expected enhancement?

Sub-experiments have to be indexed in a consistent way so that we can package them for computation (i.e. for different noise factors), and recover them in the right order for post-processing (i.e. extrapolation).

For $N$ noise factors and $M$ experiments, ZNE works by building sub-experiments for each combination of experiment and noise factor, and later analyzing the noisy results from all sub-experiments related to the same experiment to produce $M$ error mitigated results (i.e. via extrapolation). This process can be described mathematically as a mapping from $M$ experiments to $M \cdot N$ sub-experiments: $(exp_m, n) → sub_m^n$. Where $m \in [0, M[$ refers to the experiment, and $n \in [0, N[$ refers to the applied noise factor.

The question comes when we flatten this latter structure (i.e. doubly-indexed) into a single-index data type to comply with the specification for the underlying Estimator class. We call this process indexing, and it can be done in several ways with the only condition being that the process has to be reversible (i.e. multiplexing and demultiplexing) in order to group noisy results from related sub-experiments back together.

As of the current implementation, the following indexing is assumed everywhere: $(m, n) → N*m + n$. This is okay and works, nonetheless two concerns arise:

1. We would not like our solution to rely on implicit assumptions, cause it makes it hard to understand, maintain and update.
2. If at any point in time we decide to modify the indexing, we would need to do this in several places (i.e. delocalized) with a potential to introduce inconsistencies and therefore bugs.

This enhancement will consist on isolating the indexing logic within an Indexer class that will take care of the multiplexing and demultiplexing processes, effectively decoupling this logic from the rest of the code and opening up the potential for seamlessly introducing new indexing strategies in the future.