Quantum-Selected Configuration Interaction (QSCI)

Use a quantum computer only to select the most important electronic configurations, then diagonalize that small subspace classically — a hybrid scheme that asks the quantum device for the one thing it is good at and leaves the rest to proven classical linear algebra.

The problem it solves

The exact ground state of a molecule is a superposition over electronic configurations — Slater determinants, each a way of placing the electrons in the available orbitals. Full configuration interaction (full CI) keeps every determinant and diagonalizes the Hamiltonian in that complete space; it is exact, but the number of determinants grows combinatorially with the number of orbitals and electrons, so full CI hits an exponential wall after only a handful of atoms.

The saving grace is that for weakly to moderately correlated systems the ground state is often near-sparse in this basis: a small minority of configurations carry almost all of the weight, and the rest contribute next to nothing (for strongly correlated, multireference systems this sparsity is much weaker). If you could name just those dominant determinants, you could diagonalize the Hamiltonian in a tiny subspace and recover most of the correlation energy. Classical "selected CI" methods spend their effort hunting for that important set with iterative heuristics. QSCI replaces this classical heuristic hunt with a repeated sampling procedure: it gathers the high-frequency bitstrings over many shots, then filters them by physical sector — electron number, spin, symmetry — keeping only valid Slater determinants as candidate basis states for the CI subspace.

Intuition — let the quantum state vote

Prepare an approximate quantum state — a shallow ansatz, or the output of a (possibly under-converged) VQE — that already concentrates its amplitude on the chemically relevant configurations. Now measure it in the computational basis. Each measured bitstring is a configuration (which orbitals are occupied), and the configurations you see most often are exactly those with the largest amplitude. The sampler is doing importance sampling over an astronomically large configuration space for free: you never enumerate the determinants, you just let the state tell you which ones matter.

The mechanism — diagonalize in the selected subspace

Collect the most frequent bitstrings as a basis of configurations {|D₁⟩, …, |D_k⟩}. Project the molecular Hamiltonian onto that span by computing the matrix elements Hij = ⟨Di|H|Dj⟩, giving a small k×k matrix, and diagonalize it on a classical computer for its lowest eigenvalue. If the Hij are constructed exactly in one finite orbital basis and within a valid physical sector, the lowest eigenvalue of the subspace obeys the variational principle and is an upper bound on the exact ground-state energy within that finite space — a bound that can only improve as you enlarge the selected subspace.

1. Prepare an approximate quantum state (e.g. a shallow ansatz).
2. Sample it — the most frequent bitstrings name the dominant configurations.
3. Build the Hamiltonian in that selected subspace.
4. Diagonalize classically for the ground-state energy.

Seeing it in code

Read the snippet against the steps above. qalgora.sample(...) draws the bitstrings from the ansatz; is_valid_sector(...) keeps only bitstrings in the right physical sector (electron number, spin, symmetry); sorting by count and slicing the top 64 performs the selection step; build_subspace_hamiltonian(configs) assembles the k×k projected matrix; and classical_diagonalize(H_sub) is the classical eigensolve that returns the variational energy.

import qalgora
from collections import Counter

counts = qalgora.sample(ansatz, theta, shots_count=20000)
# keep only bitstrings in a valid physical sector (electron number, spin, symmetry)
valid = {b: c for b, c in counts.items() if is_valid_sector(b)}
# keep the top-k most probable configurations
top = sorted(valid.items(), key=lambda kv: -kv[1])[:64]
configs = [bits for bits, _ in top]

H_sub = build_subspace_hamiltonian(configs)   # classical
energy = classical_diagonalize(H_sub)
print("QSCI ground-state energy:", energy)

What it is — and isn't — good for

• Relatively noise-tolerant, not immune. QSCI only needs the sampler to surface the right configurations, not to reproduce their amplitudes precisely, because the energy ultimately comes from a classical diagonalization. Noise that reshuffles probabilities but still leaves the important determinants in the top-k is largely harmless — which is why QSCI runs on hardware that would derail an energy-estimating VQE. But if noise changes the ranking enough that an important configuration drops out of the selection, or it is missed altogether, accuracy still degrades.
• You can only diagonalize what you sample. If an important configuration never appears in the samples, it is absent from the subspace and its correlation energy is simply lost. The method is only as good as the state you prepare and the number of shots you can afford.
• Rare-but-important configurations are expensive. Resolving a determinant that carries small-but-nonzero weight requires many shots, so the sampling cost grows as you chase the last fractions of the correlation energy.
• The classical step must stay tractable. The selected subspace is small by construction, but computing the matrix elements and diagonalizing still bounds how large k can be.