Incorporate SQD feedback

[kaelynj]

Closes #2360

Tried to parse the feedback as best I could and would especially appreciate a review from either @garrison or @mrossinek to double check my work. Essentially all I did here was give a bit more detail on the configuration recovery step of SQD both in the intro and getting started pages.

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[qiskit-bot]

One or more of the following people are relevant to this code:

• @kaelynj

[abbycross]

Suggested change

"One important note here is how to address the very first iteration of the configuration recovery loop. Here the average orbital occupancy is not yet available and so only the `postselect_and_subsample()` method is called. This removes any non-physical samples (samples with incorrect Hamming weight) before running the eigenstate solver, `solve_fermion()`. Afterwards, the average orbital occupation is calculated across all batches and used as input to the `recover_configurations()` method which flips individual bits probabilistically based on this average. The probability of flipping an individual bit is based on the distance between the value of the bit and the average orbital occupancy obtained. (See Section **II-A** of the supplementary information in the [SQD paper](https://arxiv.org/abs/2405.05068) for more information.)"

"It is important to note how to address the first iteration of the configuration recovery loop. Since the average orbital occupancy is not yet available, only the `postselect_and_subsample()` method is called. This removes any non-physical samples (samples with incorrect Hamming weight) before running the eigenstate solver, `solve_fermion()`. Afterward, the average orbital occupation is calculated across all batches and used as input to the `recover_configurations()` method, which flips individual bits probabilistically based on this average. The probability of flipping an individual bit is based on the distance between the value of the bit and the average orbital occupancy obtained. (See Section **II-A** of the supplementary information in the [SQD paper](https://arxiv.org/abs/2405.05068) for more information.)"

[abbycross]

Suggested change

The bulk of the SQD workflow lies here wherein each of these subspace Hamiltonians is diagonalized. The ground states obtained from each of these subspaces, $|\psi^{(k)}\rangle$, are used to obtain an estimate of a reference vector of occupancies $\mathbf{n}^{(K)}$ averaged over each of the $K$ subspaces. A new set of configurations $\mathcal{X}_R$ is then generated by probabilistically flipping individual bits based on this average occupation and the known total number of particles (hamming weight) in the system. This configuration recovery process is then repeated by preparing a new set of subspaces to diagonalize, obtaining new eigenstates and averaged orbital occupancy, then generating a new set of configurations. This loop is iterated until a user specified criterion is met, and the overall process is analogous to filtering a noisy signal to improve its fidelity.

The bulk of the SQD workflow lies here wherein each of these subspace Hamiltonians is diagonalized. The ground states obtained from each of these subspaces, $|\psi^{(k)}\rangle$, are used to obtain an estimate of a reference vector of occupancies $\mathbf{n}^{(K)}$ averaged over each of the $K$ subspaces. A new set of configurations $\mathcal{X}_R$ is then generated by probabilistically flipping individual bits based on this average occupation and the known total number of particles (Hamming weight) in the system. This configuration recovery process is then repeated by preparing a new set of subspaces to diagonalize, obtaining new eigenstates and averaged orbital occupancy, and generating a new set of configurations. This loop is iterated until a user-specified criterion is met, and the overall process is analogous to filtering a noisy signal to improve its fidelity.

[abbycross]

a couple copyedit changes, otherwise LGTM!