Qiskit / qiskit-addon-sqd / 36212856842
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Loosen the energy tolerance in test_fermion.py (#377)

`test_diagonalize_fermionic_hamiltonian_basic` asserted its energy with
`places=2`, i.e. a tolerance of 0.005 Ha. That is tighter than the spread
the method actually has with `samples_per_batch=10` and `max_iterations=5`.

Scanning 200 seeds through `self.rng` in pure Python, the energy error
|result.energy + nuclear_repulsion_energy - exact_energy| ranged from
0.0014 to 0.026 Ha (median 0.0078, mean 0.0085, stdev 0.0044). The
`places=2` threshold fails for 156 of those 200 seeds; the committed seed
gives 0.0036 and so passes by luck rather than because the assertion
describes the method's accuracy. Replace it with `delta=0.05`, about
twice the observed worst case and above mean+6*sigma. `delta=` rather
than `places=1` because it states the intended tolerance directly, and
because `places` cannot express anything between 0.005 and 0.05.

The looser bound still has teeth: the SCF-to-CASCI correlation window
here is ~0.13 Ha, so it fails if SQD recovers less than ~62% of the
correlation energy, against ~93% today.

The other seeded assertions were measured and left alone. The
subspace-dimension bound `assertLess(sci_dim, 0.5 * fci_dim)` held for
200/200 seeds, though not by much: with `symmetrize_spin=True` the
subspace is square, so sci_dim only takes values k**2, and the largest
observed k=39 gives 48.5% of fci_dim while the next reachable value k=40
gives 51%. That margin is thin, but the bound is measuring something
real about this sample-and-iteration budget, so it is left as is. Note
that k=39 is not a ceiling: it is where max_iterations=5 stops the
growth, and k is still climbing when it does. Raising the iteration count
carries k well past it (k=53 at 12 iterations for one seed), with the
energy error collapsing to 6e-5 as the subspace approaches the full
space, which is SQD converging as intended.

The spin-square checks agree to at most 1.5e-14 because they compare the
same quantity comput... (continued)

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