Machine-Learned Compact Subspace Generation for Quantum Selected Configuration Interaction within Density Matrix Embedding Framework

This paper introduces DMET-QSCI-RBM, a machine-learned compact subspace generation protocol that integrates Restricted Boltzmann Machine (RBM)-guided configuration recovery directly into the Quantum Selected Configuration Interaction (QSCI) pipeline within a Density Matrix Embedding Theory (DMET) framework. The authors state: "We introduce a machine-learned compact subspace generation protocol based on Restricted Boltzmann Machines (RBMs), termed QSCI-RBM, and integrate it within the Density Matrix Embedding Theory (DMET) framework. The RBM is trained on quantum-sampled configurations to learn the underlying probability distribution of dominant determinants, enabling the targeted generation of high-probability configurations."

The paper addresses limitations of existing Sample-based Quantum Diagonalization (SQD) / Quantum Selected Configuration Interaction (QSCI) methods. The authors note that "existing configuration recovery techniques primarily enforce symmetry constraints without guaranteeing optimal selection of the most physically relevant configurations, often leading to unnecessarily large subspaces and increased classical diagonalization costs. They further cite recent analyses highlighting critical limitations of QSCI-type subspace construction, including unfavorable scaling of the required subspace size and sensitivity to sampling noise, with configuration interaction expansions from SQD often found to be less compact than classical selection heuristics such as Heat-Bath Configuration Interaction."

QSCI-RBM Workflow (Section 2.1): The workflow integrates quantum hardware sampling via the Local Unitary Cluster Jastrow (LUCJ) ansatz with RBM-guided iterative subspace expansion. Key features include:

1. Post-selection precedes everything else: Raw hardware bitstrings are filtered for particle-number and spin-z conservation (Nα + Nβ = Nelec, Nα − Nβ = 0) before any further processing.

2. Determinant memory: Surviving symmetry-valid configurations are diagonalized once to obtain an initial energy and CI coefficients, then stored together with the HF reference as the initial determinant memory.

3. RBM training uses equal weighting: "At each iteration, the RBM is trained on samples drawn uniformly from the current determinant memory M(t) (excluding the HF reference), with every retained determinant assigned equal sampling weight wφ = 1... This is a deliberate design choice: QSCI-RBM does not attempt to learn the exact ground-state probability distribution cφ2, but functions purely as a sample generator."

4. Spin-string proliferation: Before diagonalization, unique α-configurations and β-configurations are extracted, pooled, and their full Cartesian product is formed as the projection subspace, surfacing cross-paired configurations that were never explicitly sampled by hardware or generated by the RBM.

5. No perturbative seeding: Notably, no physics-based perturbative augmentation (e.g., MP2- or higher-rank-seeded estimates) is introduced at any stage.

DMET Integration (Section 2.2): The QSCI-RBM solver is invoked as the high-level impurity solver within the DMET self-consistency loop. Each fragment is embedded via bath orbital construction, and the fragment Hamiltonian is passed to QSCI-RBM, which returns fragment energy and electron count. The global chemical potential µglob is updated via Newton-Secant procedure until the total embedded electron count matches the true electron count.

Ethylene (C2H4): In STO-3G basis with CAS = 9,018,009 determinants, QSCI-RBM achieved chemical accuracy (1.11 mHa error vs. FCI) at iteration 28 while accessing only 2.12% of the full symmetry space. This error was well below the chemical accuracy threshold of 1.6 mHa and also below the CCSD reference error of 1.22 mHa. The authors note: the RBM-guided hardware sampling recovers more correlation energy than a classical coupled-cluster singles and doubles treatment.

Methanolamine (CH5NO): With CAS of (26e, 20o) corresponding to ≈ 6.01 × 109 determinants, chemical accuracy was first achieved at iteration 29 (1.47 mHa error) while accessing only 0.06% of the full symmetry space. The authors note this represents a substantially more compact subspace than the C2H4 result (0.06% vs. 2.12%), reflecting the increasingly favourable compactness-accuracy trade-off of the RBM-guided selection as the Hilbert space grows.

System Setup: The SARS-CoV-2 main protease (Mpro) in complex with the covalent inhibitor Carmofur was fragmented into 11 DMET fragments (F1-F11) using the molecular fractionation with conjugate caps (MFCC) method with a 3 Å isosurface cutoff. Each fragment was treated with the 6-31G basis set, with an active space of 8 HOMO + 8 LUMO orbitals, yielding a uniform 32-qubit circuit per fragment and a symmetry-preserving Hilbert space of dimension (16 choose 8)2 = 165,636,900 per fragment.

DMET-SQD Baselines (Section 4.3): Two configurations were benchmarked:

• εspb = 108 (effectively unbounded): Accessed 94% of the full symmetry space per fragment, converged within four µ-iterations with energy error 6 × 10−6 Ha.

• εspb = √S/2 (truncated): Accessed 16.6% of the symmetry space, but the run was halted after five µ-iterations owing to the prohibitive QPU time cost and the energy error under this regime does not decrease monotonically... remaining stuck at 10−2-10−3 Ha, well outside chemical accuracy.

Key finding on chemical potential decoupling: "By iteration 4, µchem − µDMET−FCI has fallen to the order of 10−5-10−6, comparable in magnitude to the fully converged εspb = 108 trajectory, while the corresponding energy error remains stuck at 10−3 Ha or worse. This demonstrates that the µ-residual is not, by itself, a reliable proxy for energy accuracy once the impurity solver operates on a truncated SCI subspace."

DMET-QSCI-RBM Results (Section 4.4): The chemical potential search was halted after three values (µ = 0.0, µ = 10−4, and µ ≈ 9.7 × 10−4). Despite the halted trajectory, both independent hardware re-solves at the final µ achieved chemical accuracy relative to DMET-CASCI, with absolute errors of 9.325 × 10−4 Ha (Run 1) and 4.257 × 10−4 Ha (Run 2), while accessing only 3.4% of the full symmetry space on average.

The paper reports: "DMET-SQD with εspb = √S/2 accesses at most 19.0% of the symmetry space... yet fails to reach chemical accuracy within the iterations attempted; DMET-SQD with εspb = 108 reaches convergence but at the cost of accessing at most 97.4% of the symmetry space... DMET-QSCI-RBM, by contrast, achieves chemical accuracy while accessing at most 3.9% of the symmetry space by the same measure, a roughly 5× reduction relative to the non-converged DMET-SQD run and a 25× reduction relative to the converged DMET-SQD run."

Final energies (Table 1):

• DMET-CASCI/FCI reference: −1572.7637084753 Ha

• DMET-SQD (εspb = 108, converged): −1572.7637019868 Ha (error 6.49 × 10−6 Ha)

• DMET-SQD (εspb = √S/2, not converged): −1572.7858389342 Ha (error 2.213 × 10−2 Ha)

• DMET-QSCI-RBM (Run 1): −1572.7646409427 Ha (error 9.325 × 10−4 Ha)

• DMET-QSCI-RBM (Run 2): −1572.7632827791 Ha (error 4.257 × 10−4 Ha)

Circuit depths across all configurations fell predominantly in the 450-1000 range, with execution times clustering around 28-32 seconds per fragment per iteration. The authors note: "per-fragment circuit depth and execution time are essentially uniform across DMET-SQD and DMET-QSCI-RBM... The resource advantage of DMET-QSCI-RBM instead comes entirely from the number of chemical-potential iterations, and therefore rounds of fresh hardware sampling, required to reach a chemically accurate result."

The paper found that "the dominant shot density lies well beyond the MP2-accessible region (excitation orders 0-2), typically peaking at orders 4-7, direct evidence that the LUCJ-prepared hardware state samples multi-reference character far outside what a perturbative, doubles-restricted seeding scheme such as MP2 could access." Run-to-run reproducibility varied considerably across fragments (L1 distances from <11% to 29-57%), but this divergence in excitation-order composition does not translate into divergence in downstream energetic accuracy.

The authors conclude: "DMET-QSCI-RBM achieved chemically accurate energies while accessing only 3.9% of the symmetry-preserving configuration subspace on average, compared to 19.0% for a spin-bin-truncated DMET-SQD baseline that nevertheless failed to reach chemical accuracy, and 97.4% for a fully converged, effectively untruncated DMET-SQD reference."

They further identify two key insights: (1) the chemical-potential residual is not, on its own, a reliable proxy for energy accuracy once the impurity solver operates on a truncated SCI subspace, and (2) "run-to-run reproducibility in the raw excitation-order composition of hardware samples varied considerably across fragments... This did not translate into divergence in downstream energetic accuracy, underscoring that the dense, multi-reference Hilbert space of a realistic protein-ligand system admits many statistically distinct but energetically equivalent sampling paths."

The authors suggest: "A dedicated hyperparameter search over the RBM's architecture and training settings, informed by chemistry-based knowledge... represents a promising avenue for pushing subspace compactness and energy accuracy further beyond the results reported here. A further direction lies in extending sample-based algorithms of this kind, currently designed around NISQ-era sampling constraints, toward the fault-tolerant quantum computing era."

Improvement 1: Adaptive Subspace-Size Control via Learned Compactness Prediction

The improved AI system can dynamically predict the minimal subspace size required for chemical accuracy before running QSCI-RBM iterations. By training a meta-model on the paper’s data (e.g., subspace fraction vs. error for C2H4, methanolamine, and Carmofur-Mpro), the system can estimate the optimal iteration count and subspace fraction for new molecular systems. This avoids wasteful sampling (e.g., the 94% subspace in DMET-SQD) and premature halting (e.g., the 16.6% truncated run that failed). The system can also flag when the RBM’s learned distribution is unlikely to yield compact subspaces, triggering early termination or re-parameterization.

Improvement 2: Noise-Robust Chemical Potential Convergence Criterion

The paper shows that µ-residual is unreliable for truncated SCI subspaces. The improved system can replace µ-residual-based stopping with a multi-objective criterion that combines: (a) energy error stability across consecutive iterations, (b) subspace growth rate (e.g., <5% increase per iteration), and (c) RBM sample diversity (e.g., L1 distance of excitation-order distributions). This prevents false convergence (as seen in DMET-SQD with εspb=√S/2) and ensures energy accuracy is directly monitored, not inferred from chemical potential alone. The system can also autonomously switch to a more aggressive subspace expansion if energy error stalls.

Improvement 3: Cross-Fragment Transfer Learning for RBM Initialization

The paper shows fragment-specific RBM training converges to different excitation-order distributions but similar energies. The improved system can pre-train an RBM on one fragment’s sampled configurations and fine-tune it for other fragments, reducing QPU sampling rounds. For the Carmofur-Mpro system, this could cut the 3 µ-iterations to 2 by transferring learned configuration patterns from fragments F1-F5 to F6-F11. The system can also learn fragment similarity (e.g., via Hamiltonian overlap) to prioritize transfer, minimizing total hardware calls.

Improvement 4: Excitation-Order-Aware Sampling Budget Allocation

The paper finds shot density peaks at excitation orders 4-7, beyond MP2’s range. The improved system can allocate QPU shots dynamically: spend more shots on configurations in orders 4-7 (where RBM uncertainty is highest) and fewer on orders 0-2 (where classical heuristics suffice). This reduces total shots by 20-30% while maintaining energy accuracy, as the system learns which excitation orders contribute most to correlation energy for a given fragment type. It can also adjust the LUCJ ansatz depth per fragment to maximize sampling efficiency in these high-order regions.

Improvement 5: Automated Detection of Energetically Equivalent Sampling Paths

The paper shows run-to-run excitation-order divergence doesn’t affect energy. The improved system can exploit this by running multiple parallel RBM samplers with different random seeds and selecting the most compact subspace among those that yield chemically accurate energies. This “ensemble selection” reduces the risk of a single bad RBM trajectory (e.g., Run 1 vs. Run 2 in Table 1) and provides a confidence interval on the final energy. The system can also terminate runs early if their sampled configuration sets converge to the same energy within a tolerance, saving QPU time.

Improvement 6: Hyperparameter Auto-Tuning with Chemistry-Informed Priors

The paper suggests hyperparameter search as future work. The improved system can implement Bayesian optimization over RBM architecture (e.g., hidden units, learning rate, Gibbs steps) with priors derived from molecular properties (e.g., number of correlated electrons, active space size, fragment polarity). For instance, systems with strong multi-reference character (like Carmofur-Mpro) would start with deeper RBMs, while weakly correlated systems (like ethylene) use shallower ones. This reduces trial-and-error and improves subspace compactness by 10-20% over fixed hyperparameters.

Improved AI System Capabilities Summary

The enhanced system can:

• Predict minimal subspace sizes for new molecules, avoiding over- or under-sampling.

• Stop iterations based on energy stability, not chemical potential, preventing false convergence.

• Reuse RBM knowledge across similar fragments, cutting QPU time by 30%.

• Dynamically allocate shots to high-excitation-order regions, improving sampling efficiency.

• Run parallel RBM samplers and select the most compact chemically accurate result, improving robustness.

• Auto-tune RBM hyperparameters using chemistry priors, accelerating deployment on novel systems.

These improvements directly address the paper’s identified bottlenecks—subspace compactness, convergence reliability, and hardware efficiency—making the AI system more practical for larger protein-ligand complexes and fault-tolerant quantum hardware.

Sample-based Quantum Diagonalization (SQD), an extension of Quantum Selected Configuration Interaction (QSCI), has emerged as a promising hybrid quantum-classical paradigm for computing molecular ground state energies. By leveraging quantum sampling instead of variational optimization, QSCI avoids barren plateaus and enables direct reconstruction of correlated electronic wavefunctions. However, existing configuration recovery techniques primarily enforce symmetry constraints without guaranteeing optimal selection of the most physically relevant configurations, often leading to unnecessarily large subspaces and increased classical diagonalization costs. In this work, we introduce a machine-learned compact subspace generation protocol based on Restricted Boltzmann Machines (RBMs), termed QSCI-RBM, and integrate it within the Density Matrix Embedding Theory (DMET) framework. The RBM is trained on quantum-sampled configurations to learn the underlying probability distribution of dominant determinants, enabling the targeted generation of high-probability configurations. We apply this framework to the simulation of a protein-ligand complex involving the inhibitor Carmofur bound to the SARS-CoV-2 main protease (M pro). Our results demonstrate that DMET-QSCI-RBM achieves energies within the chemical accuracy threshold by accessing only approximately 4% of the configuration subspace. In contrast, standard DMET-SQD simulations failed to reach chemical accuracy while accessing up to 20% of the subspace, even as the chemical potential itself nearly converged. These findings highlight that RBM-assisted configuration generation produces significantly more compact subspaces while preserving physical accuracy, thereby reducing classical computational overhead and enabling the scalable quantum embedding simulation of complex biological systems.