Quantum proposals lower energy in H12 molecule at stretched geometry

Researchers at Quantinuum K.K. and NVIDIA Corporation demonstrated a new hybrid quantum-classical method that lowers the variational energy for calculations on the H12 molecule at a “stretched geometry”. The work proposes a system where quantum measurements suggest determinants only accepted if they energetically “outrank” classical candidates, all while maintaining a fixed computational budget. This approach targets multistep excitations often missed by purely classical methods used in electronic-structure calculations. The demonstration used the Quantinuum Reimei trapped-ion quantum computer to supply complementary determinants, improving selected configuration interaction accuracy.

Quantum Residual Exploration for Selected Configuration Interaction

Researchers at Quantinuum K.K. and NVIDIA Corporation developed a hybrid quantum-classical method that improves selected configuration interaction (SCI) while maintaining a fixed computational budget for final determinants. This approach accepts quantum-suggested determinants only if they energetically “outrank” those identified through purely classical methods, a stringent criterion designed to maximize efficiency.

The new method addresses a limitation of classical SCI, which can struggle to identify important determinants in strongly correlated systems due to the Hamiltonian’s coupling of determinants through multiple substitutions. Classical methods often rely on local importance estimates for determinant discovery, a process that can become inefficient when resources are limited; the team’s approach aims to overcome this by using quantum measurements to propose determinants that might otherwise be missed.

A key aspect of the work is the use of heat-bath configuration interaction (HCI) as a reproducible classical baseline for comparison, allowing for a clear assessment of the quantum component’s contribution. The researchers constructed a system where the final determinant space is fixed, meaning the total number of determinants considered remains constant regardless of whether they originate from classical or quantum sources. This modular hybrid approach differs from previous quantum-selected configuration interaction (QSCI) methods by focusing on complementing, rather than replacing, classical techniques.

The team’s system generates a pool of candidate determinants both classically and through quantum measurements, then selects the most energetically favorable ones to populate the final variational space. “Quantum measurements supply complementary determinants, accepted only when they outrank classical candidates under a common energetic criterion,” the authors state in their report, highlighting the method’s selective acceptance process.

The demonstration on H12 at a stretched geometry showed that incorporating determinants identified through quantum measurement lowered the overall variational energy compared to relying solely on classical selection, suggesting a tangible benefit from this hybrid strategy. William Dawson of RIKEN Center for Computational Science, along with colleagues, contributed to the research detailed in the report.

H12 Molecular Geometry Impacts on Variational Energy

The inclusion of quantum-derived determinants demonstrably lowers the variational energy of the H12 molecule when stretched to 1.8 Ångströms, a finding that suggests improved accuracy in modeling molecular behavior under stress. This improvement stems from the method’s ability to identify and incorporate determinants often missed by traditional perturbative approaches, particularly those governing multistep excitations in strongly correlated systems. The research details a system where a classical electronic Hamiltonian is constructed from molecular inputs like nuclear geometry and basis sets, defining the space of possible determinants.

A key element is the evaluation of a second-order perturbative energy correction, calculated using the final selected determinants and eigenvectors, to refine the baseline energy. This correction, applied to both classical and quantum-enhanced calculations, allows for a direct comparison of their performance. The team assessed these corrections across seven different molecular geometries, demonstrating consistent energy lowering with the inclusion of quantum-suggested determinants.

The process relies on a fixed final determinant space, ensuring a consistent computational cost across all calculations. The baseline determinant pool is then augmented with determinants proposed by quantum measurements, but only those that energetically “outrank” the classical candidates are accepted. This selective acceptance, combined with a final variational diagonalization, yields the lowest singlet state energy for each geometry. The largest energy lowering from this quantum-classical refinement reached 0.44 milli-Hartree at 1.7 Ångströms when using a purely classical energy normalization, but expanded to 10.80 milli-Hartree at 1.8 Ångströms with the quantum residual exploration.

Quantum Proposal Circuit Design for Determinant Discovery

Separating proposal generation from classical energetic assessment in the design of quantum circuits to identify determinants for electronic structure calculations allows for targeted exploration of configurations often missed by traditional methods. This approach constructs a classical reference state from a complete-active-space CI calculation, embedding it within the target orbital space to initialize the quantum circuit and provide a baseline for comparison. Researchers then use this reference, alongside classical importance estimates, to select excitation pathways for coherent evolution within the quantum system.

A mid-circuit measurement rejects configurations that maintain the initial occupation pattern on frozen orbitals, effectively focusing the quantum search on potentially relevant determinants. These quantum-proposed candidates undergo processing to ensure symmetry validity and spin completeness before being combined with classically generated proposals for a unified energetic ranking.

The process deliberately avoids multiple intermediate diagonalizations by employing a single stage of classical HCI, streamlining the computational cost while still allowing for a robust comparison of quantum and classical determinant discovery. The baseline determinant pool size varied between 16,297 and 5,537 additions across different molecular geometries, confirming this work focuses on determinant discovery rather than convergence. This methodology addresses a key limitation of classical selected configuration interaction (SCI), which can struggle to efficiently find important determinants lacking direct connections to the initial selected space.

By using quantum sampling, as proposed by Kanno et al., the researchers aim to augment classical methods with a means of discovering configurations that would otherwise require extensive, and potentially inefficient, classical searching within a fixed computational budget. The implementation details and intermediate benchmark data are available in the Supplemental Material, Section S1, providing a transparent account of the methodology and results.

Heat-Bath Configuration Interaction as a Classical Baseline

Heat-bath configuration interaction (HCI) served as the reproducible classical baseline for assessing the impact of quantum proposals, offering an efficient and explicit determinant-selection criterion. The baseline began with a truncated centered CAS(8,8) ground state, retaining at least 99% of its probability weight before embedding it within a CAS(12,12) space, a process designed to create a well-defined starting point for analysis.

This initial configuration expanded from 3,703 to 14,463 determinants across a scan at an energy threshold of 2.3 milli-hartree, demonstrating the growth of the determinant space as accuracy demands increased. The methodology deliberately contrasts with approaches like sample-based quantum diagonalization, which have shown substantial repeated-sampling overhead or required larger CI expansions for comparable energy accuracy.

This focus on minimizing overhead is motivated by prior work from Reinholdt et al., highlighting the need for efficient quantum sampling strategies. The authors retained responsibility for the scientific content and conclusions, ensuring the rigor of the methodology. The selection process, detailed in the Supplemental Material, involved constructing both classical and quantum proposal pools before applying a selection criterion to build the final CI space. This approach allows for a direct comparison of determinant discovery methods, evaluating the utility of quantum sampling as a complementary source of configurations.

Quantum vs. Classical Determinant Ranking Criteria

The methodology employed a fixed final determinant budget, allowing for a direct assessment of quantum proposals alongside classical candidates within a constrained computational space. This approach deliberately separates the generation of potential determinants from their evaluation, streamlining the process and enabling focused comparison of the two sources. Measured proposals for H12 at a stretched geometry lower the variational energy relative to classical-only selection at the same final determinant budget.

Each remaining occupation was assigned its total probability, completed with all compatible spin configurations, and admitted in descending probability order up to the capacity limit. “These probabilities determine pool construction only; final admission is decided by EN or CEN2 ranking,” the researchers state, emphasizing the rigorous selection process.

Incremental Energy Improvement via Quantum Proposals

Quantum residual exploration (QRE) introduces a method for discovering excitation pathways that classical selected configuration interaction (SCI) may require multiple iterations to reach, offering a potential refinement to established electronic structure calculations. 8 Å yielded lower variational energies compared to HCI+CEN2, all while maintaining a fixed final determinant budget.

Reference amplitudes guide the selection of transport generators, and baseline coefficients alongside energy values determine the final energetic ranking. This flexibility offers a pathway to maximize energy improvement within fixed gate and measurement budgets, even as QRE still relies on classical resources for storage and diagonalization of the selected determinant space.

Hamiltonian Construction from Molecular Inputs

Electronic Hamiltonian construction begins with molecular inputs defining the second-quantized electronic Hamiltonian, H, and classical preprocessing evaluates one- and two-body integrals. The research team applied this process to an equally spaced linear chain of H12 molecules using the STO-3G basis, correlating all 12 electrons and 12 spatial orbitals within a fixed-M_S = 0 determinant basis. This setup allows for direct comparison with exact reference energies obtained from full configuration interaction (FCI) within the (12,12) space, providing a stringent benchmark for the new method.

The workflow incorporates a perturbative energy correction, EN-PT2, to assess the final N = 20,000 HCI+CEN2 and QRE+CEN2 states across seven different molecular geometries. This correction, calculated as $E^2$ = sum_(I∉$V_{final})($|sum_(J ∈ V_(final))H_(IJ)C_J^(var)|^2)/(E_(var)-H_(II)) + $E_{var}$, uses the final eigenvector rather than initial HCI baseline or CEN2 ranking amplitudes.

The researchers emphasize that comparisons are made with a fixed final determinant dimension, prioritizing the exploration of method efficacy rather than a direct match in total computational costs. Section S2 further examines additional HCI discovery methods, while the End Matter provides a detailed account of the hardware methods and analysis employed in this study, demonstrating a commitment to transparency and reproducibility.

Limitations of Perturbative Determinant Selection

This sensitivity highlights a limitation; a poorly chosen initial threshold could hinder the discovery of beneficial determinants, even with quantum assistance. This design choice means the method’s practical utility for high-precision calculations remains an open question, and further work is needed to demonstrate its ability to reach stringent accuracy thresholds.

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