Enhancing quantum-classical configuration interaction methods using a neural-network classifier

Technical Summary: Enhancing Quantum-Classical Configuration Interaction Methods Using a Neural-Network Classifier

Problem Statement
Accurate electronic structure calculations require methods capable of handling electron correlation within Hilbert spaces that grow exponentially with the number of active orbitals. While Full Configuration Interaction (FCI) provides exact solutions, its computational cost restricts it to very small systems. Selected Configuration Interaction (SCI) methods, such as CIPSI, ASCI, and Heat-Bath Configuration Interaction (HCI), offer a compromise by iteratively constructing compact variational spaces containing only the most important determinants. However, the efficiency of SCI methods relies critically on heuristics used to identify these important determinants. As the selected space grows, the number of candidate determinants to inspect becomes large, creating a bottleneck.

Similarly, in the quantum computing domain, Sample-based Quantum Diagonalisation (SQD) methods attempt to solve the CI problem using NISQ devices by constructing a subspace from quantum measurements. A significant limitation of SQD is its inefficiency in discovering new determinants; repeated sampling often selects the same configurations, particularly when starting from an approximate ansatz, which limits calculation accuracy.

Methodology
The authors propose a data-driven framework that recasts determinant selection as a binary classification task, integrating a neural-network (NN) classifier into iterative CI workflows via an active-learning loop. This approach is applied to two distinct algorithms: a classical variant of HCI (cHCI) and a quantum-classical variant of SQD (cSQD).

The core procedure operates as follows at each iteration:

1. Candidate Pool Generation: The underlying method generates a pool ($P$) of candidate determinants (e.g., via heat-bath screening in HCI or configuration recovery in SQD).
2. Random Sampling and Labeling: A small, fixed fraction ($f_s = 0.2$) of the pool is randomly selected ($P_{rand}$). A temporary variational space is formed by combining the current space with $P_{rand}$. A single diagonalization of the Hamiltonian in this temporary space yields CI coefficients.
3. Label Assignment: Configurations in $P_{rand}$ with amplitudes exceeding a threshold are labeled "important," while others are labeled "unimportant." This creates a balanced dataset for training.
4. Classifier Training: A neural-network classifier is trained from scratch on this labeled subset to predict the importance of configurations based on their occupation-number bitstrings.
5. Guided Selection: The trained classifier scores the remaining candidates in the pool ($P_{rest}$). The next variational space is constructed by retaining the labeled "important" configurations from $P_{rand}$ and adding the top-scoring configurations from $P_{rest}$.

This framework modifies only the selection step of the algorithm, leaving the core workflow (diagonalization, sampling, etc.) intact. The classifier adapts continuously as the wave function evolves during the iterative process.

Key Contributions

• Unified Framework: The paper introduces a general active-learning loop that treats determinant selection as a binary classification problem, applicable to both classical and quantum-classical SCI methods.
• Algorithmic Integration: The authors successfully integrate this framework into two specific algorithms:
  • cHCI: A classifier-enhanced Heat-Bath Configuration Interaction method.
  • cSQD: A classifier-enhanced Sample-based Quantum Diagonalisation method.
• Bitstring-Based Classification: The approach utilizes neural networks that operate directly on occupation-number bitstrings, providing a flexible, architecture-agnostic alternative to traditional perturbative or energy-based selection criteria.

Results
The authors benchmarked the proposed methods against their traditional counterparts using the ground state of the nitrogen molecule ($N_2$) at an equilibrium internuclear distance of $R = 2.1$ Å in the cc-pVDZ basis set.

• cHCI Performance: The cHCI method recovered more than 98% of the CASCI correlation energy using a variational space dimension that was only 1/5 the size of the converged standard HCI space. This resulted in a roughly 5-fold reduction in both memory footprint and per-iteration computational cost (specifically for the variational Davidson diagonalization).
• cSQD Performance: The cSQD method demonstrated significantly accelerated convergence compared to standard SQD. cSQD achieved a relative correlation-energy error below 1% in just 3 iterations. In contrast, standard SQD failed to reach this accuracy threshold even after 20 iterations.
• Efficiency: The classifier-based selection effectively compressed the variational space without sacrificing accuracy, allowing the algorithms to focus computational effort on the most relevant configurations.

Significance and Claims
The paper claims that classifier-assisted determinant selection serves as a lightweight, method-agnostic tool for compressing variational spaces and accelerating both classical and hybrid quantum-classical configuration interaction algorithms. By replacing heuristic selection with a data-driven classifier trained on-the-fly, the method achieves result parity with traditional approaches at substantially lower computational costs.

The authors position this work as an initial step toward a more comprehensive understanding of classifier-assisted selection. They note that while the current study focuses on a single molecular system and the variational stage, the framework suggests a broader role for machine learning in designing compact wave function representations and more efficient electronic structure workflows. The paper does not claim to solve the scaling problem for all systems universally but demonstrates the utility of the approach for the specific cases of cHCI and cSQD on $N_2$.