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

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

Problem Statement
Accurate simulation of correlated electronic structures in large molecular systems, particularly those of pharmaceutical relevance, remains a significant challenge. Exact diagonalization methods like Full Configuration Interaction (FCI) scale combinatorially, rendering them intractable for moderate-to-large systems. While hybrid quantum-classical algorithms like Variational Quantum Eigensolvers (VQE) offer a pathway, they suffer from optimization pathologies such as barren plateaus and noise sensitivity. Sample-based Quantum Diagonalization (SQD), or Quantum Selected Configuration Interaction (QSCI), addresses these issues by using quantum hardware solely for sampling bitstrings rather than iterative optimization, with classical post-processing performing exact diagonalization on the sampled subspace. However, existing QSCI approaches face a critical bottleneck: the quality and compactness of the sampled configuration subspace. Raw hardware samples are noisy and often violate physical symmetries (e.g., electron number, spin). Current recovery techniques project these samples back onto valid symmetry sectors but do not guarantee the selection of the most physically relevant configurations. This often results in unnecessarily large subspaces that increase classical diagonalization costs and erode potential quantum advantages. Furthermore, applying QSCI to large biological systems requires fragmentation strategies like Density Matrix Embedding Theory (DMET), which introduces additional complexity in managing the trade-off between subspace size and chemical accuracy.

Methodology
The authors propose DMET-QSCI-RBM, a unified framework that integrates a machine-learned compact subspace generation protocol into the DMET workflow. The core innovation is the use of a Restricted Boltzmann Machine (RBM) trained directly on quantum-sampled bitstrings to learn the probability distribution of dominant determinants, thereby enabling the targeted generation of high-probability configurations without relying on perturbative seeding (e.g., MP2).

The methodology proceeds through the following key steps:

1. Quantum Sampling: A Local Unitary Cluster Jastrow (LUCJ) ansatz is executed on quantum hardware to generate raw bitstrings.
2. Post-Selection and Initialization: Raw samples are filtered for particle-number and spin conservation. The surviving configurations, along with a Hartree-Fock (HF) reference, form the initial "determinant memory."
3. RBM-Guided Iterative Expansion:
   • The RBM is trained on the current determinant memory (excluding the HF reference) using uniform weights, rather than weighting by CI coefficients, to ensure unbiased exploration of the dominant sector.
   • The trained RBM performs Gibbs sampling to generate new candidate determinants.
   • These candidates are filtered for symmetry and novelty (removing those already in memory).
   • The filtered candidates are unioned with the current memory to form a working diagonalization space.
4. Spin-String Proliferation: Before diagonalization, unique $\alpha$ and $\beta$ configurations are extracted and their Cartesian product is formed. This step surfaces cross-paired configurations not explicitly sampled, expanding the subspace without additional quantum shots.
5. Classical Diagonalization and Memory Update: The projected Hamiltonian is diagonalized classically (using the Davidson method). Configurations with CI coefficients exceeding a threshold are permanently added to the determinant memory for the next iteration.
6. DMET Integration: This QSCI-RBM solver acts as the high-level impurity solver within the DMET self-consistency loop. The loop adjusts the global chemical potential ($\mu_{glob}$) until the total embedded electron count matches the true electron count.

Key Contributions

• RBM-Driven Subspace Construction: The paper introduces a method to replace or augment traditional configuration recovery heuristics with a learned generative model. This allows for the direct targeting of compact, high-quality determinant subspaces without perturbative anchoring.
• Integration with DMET: The framework successfully couples QSCI-RBM with DMET, enabling the simulation of large, fragmented biological systems while keeping the qubit count per fragment constant.
• Novelty and Symmetry Filtering: The workflow includes specific mechanisms to prevent double-counting and ensure symmetry validity during the iterative expansion of the subspace.
• Spin-String Proliferation: The authors utilize a proliferation step to implicitly expand the subspace using the Cartesian product of unique spin strings, capturing correlations missed by direct sampling.

Results
The framework was validated on small molecules (ethylene and methanolamine) and applied to a complex biological system: the Carmofur inhibitor bound to the SARS-CoV-2 main protease (Mpro).

• Small Molecule Benchmarks: On ethylene ($C_2H_4$) and methanolamine ($CH_5NO$), DMET-QSCI-RBM achieved chemical accuracy (error < 1.6 mHa) while accessing only 2.12% and 0.06% of the full symmetry-adapted configuration space, respectively. These results outperformed or matched classical CCSD benchmarks while using significantly smaller subspaces.
• Protein-Ligand Simulation (Carmofur-Mpro):
  • The system was fragmented into 11 DMET impurities, each with a uniform (16e, 16o) active space (32 qubits).
  • DMET-SQD Baselines: A standard DMET-SQD approach with an unbounded subspace ($\epsilon_{spb} = 10^8$) reached chemical accuracy but required accessing ~97.4% of the symmetry space. A truncated baseline ($\epsilon_{spb} = \sqrt{|S|/2}$) accessed only ~19.0% of the space but failed to reach chemical accuracy, with energy errors remaining around $10^{-2}$ Ha. Notably, the chemical potential residual converged for the truncated baseline, yet the energy did not, highlighting that $\mu$-convergence is not a reliable proxy for energy accuracy in truncated subspaces.
  • DMET-QSCI-RBM Performance: The proposed method achieved chemical accuracy (errors of $9.3 \times 10^{-4}$ Ha and $4.3 \times 10^{-4}$ Ha in two independent runs) while accessing only ~3.9% of the symmetry space on average. This represents a roughly 25-fold reduction in subspace size compared to the converged DMET-SQD baseline and a 5-fold reduction compared to the non-converged truncated baseline.
  • Efficiency: The method achieved these results with fewer chemical-potential iterations, reducing the total quantum sampling cost. The quantum circuit depth and execution time per fragment were comparable across all methods, indicating that the efficiency gain stems from the reduced number of iterations required to reach convergence.

Significance and Claims
The paper claims that DMET-QSCI-RBM establishes subspace compactness as a key practical advantage of learned, hardware-driven configuration selection. The authors assert that:

1. Compactness vs. Accuracy: Machine-learned subspace generation produces significantly more compact subspaces than standard configuration recovery or perturbative seeding while preserving physical accuracy.
2. Decoupling of Metrics: Energy accuracy in DMET-QSCI-RBM is governed primarily by the quality and compactness of the accumulated determinant memory, not by the degree of chemical potential self-consistency. This allows the framework to reach chemically meaningful results even when the $\mu$-search is halted early.
3. Scalability: By reducing the classical diagonalization cost per fragment, this approach enables the scalable quantum embedding simulation of complex biological systems (like protein-ligand complexes) that would otherwise be intractable due to the exponential scaling of the configuration space.
4. Hardware Agnosticism: The method leverages the specific capabilities of NISQ-era sampling (avoiding barren plateaus) while mitigating the noise and subspace size issues inherent to raw sampling through learned generative models.

The authors conclude that this approach paves the way for classically tractable quantum simulations of biomolecular systems in their full pharmacological context, moving beyond the limitations of truncated active-site models. They note that further improvements could be realized through systematic hyperparameter optimization of the RBM and comparative studies with other compact subspace strategies.