BQM should store the lowest known energy

[arcondello]

It might be useful for samplers to be able to "annotate" binary quadratic models with data such as the lowest energy found by any solver and/or the state(s) that found that energy. This lowest energy could potentially come with a certificate marking it as a ground state (when that's possible).

Uses

• The current best state(s) could be used to see other solvers that accept an initial state (see also Consistent handling of initial states in Samplers/Composites #491)
• Useful for benchmarking, because it provides a target and different samplers could be rated on time-to-target

Additional Context
This data could just be stored in the BQM.info dict.

[joseppinilla]

I stumbled across this while looking into alternatives to store BQM information now that the info parameter will be deprecated.

What would be the recommended practice for that right now?

I'm using info to store:

'E0' : <float> # Ground State if known
'energy' : [ ... ] # sorted list of known energies
'degeneracy' : [ ... ] # sorted list of known degeneracies

I found this to be pretty thorough in the information I needed.

If there's a known "minimum" energy it can be put in the 'energy' list but not in E0, and if the degeneracy is not know, the list can be left empty.

[arcondello]

Would like to know more about where you use that information in a flow.

I think the "intended" place is alongside the BQM, e.g.

model = dict(bqm=bqm, E0=1.0, energy=[1,2], ...)

Another approach would be to add them as attributes

bqm = dimod.AdjVectorBQM('SPIN')
bqm.E0 = 1

Or be to store some of that in the sampleset, e.g.

sampleset = dimod.SampleSet.from_samples([[0, 1], [1, 0]], 'SPIN', energy=[0, 1], known_ground=[True, False])

[joseppinilla]

I use that information for benchmarking.
The energies and degeneracies would be known before sampling, which is why I add them to the BQM. Then, with that information, it is possible to build a "Boltzmann Fit" of the sampleset:

I think a good example is Fig. 16 of: https://arxiv.org/pdf/1810.05881.pdf