Hamiltonian Simulation-Based QSCI

• HSB-QSCI is a quantum-classical hybrid method that uses real-time Hamiltonian simulation to generate systematically selected configuration interaction bases with explicit size consistency.
• By integrating quantum hardware sampling with classical CI diagonalization, the method accurately computes intermolecular energies, achieving chemical accuracy in benchmark tests.
• Its structured determinant sampling, symmetry completion, and dimer-monomer subspace augmentation make it scalable for large supramolecular electronic structure calculations.

Hamiltonian Simulation-Based Quantum-Selected Configuration Interaction (HSB-QSCI) is an advanced quantum-classical hybrid methodology designed for large-scale electronic structure calculations, with a particular emphasis on obtaining accurate intermolecular interaction energies in quantum chemistry. The method fundamentally augments configuration interaction (CI) by using quantum hardware to generate compact, systematically selected basis sets via real-time quantum Hamiltonian simulation, and it incorporates explicit size consistency—an essential property for the supramolecular approach—by carefully constructing and augmenting the subspace of determinants sampled from quantum evolution.

1. Size Consistency in HSB-QSCI

Size consistency is the property that, for noninteracting fragments, the electronic energy of the combined system equals the sum of the energies of the fragments. It is formally required that

$E_{\text{AB}} = E_{\text{A}} + E_{\text{B}}$

when systems A and B are infinitely separated. In quantum chemistry, especially when estimating weak intermolecular interactions (e.g., hydrogen bonds, van der Waals complexes), lack of size consistency artificially distorts interaction energies, undermining the reliability of the supramolecular approach: $$E_{\text{int}} = E_{\text{AB}} - E_{\text{A}} - E_{\text{B}}$$ The naive implementation of QSCI lacks size consistency since the subspace of sampled Slater determinants for the dimer is not guaranteed to encompass all necessary product determinants of the monomer subspaces. The sc-HSB-QSCI (size-consistent HSB-QSCI) framework addresses this by constructing dimer and monomer subspaces in the localized orbital basis and augmenting them to ensure completeness for noninteracting cases. This augmentation involves, for any sampled dimer determinant that is a product of monomer determinants, including the monomer determinants in their respective subspaces. For fully noninteracting fragments, this restores the required additivity of energies to machine precision.

2. Determinant Sampling and Subspace Construction

The method begins by preparing a simple approximate state (typically the Hartree–Fock wavefunction) for the dimer and evolving it in time under the molecular Hamiltonian: $|\Psi_k\rangle = e^{-iH k\Delta t}|\Psi_0\rangle$ where $H$ is encoded (e.g., in the Jordan–Wigner basis) as a weighted sum of Pauli operators, and the evolution is implemented with a Trotter–Suzuki decomposition. For each time step $k\Delta t$ (with $k = 1, 2, ..., K$ and fixed $\Delta t$), thousands of measurements are performed in the computational (Z) basis.

Measured bitstrings correspond to Slater determinants. The simulation is performed in a localized molecular orbital basis (e.g., from Pipek–Mezey localization) to allow for the unambiguous assignment of determinants to individual monomers. Each sampled bitstring is categorized as:

• Intra-monomer configuration: excitations only within a single monomer. Such determinants are split between the A and B monomer subspaces.
• Inter-monomer (charge transfer): excitations spanning both monomers—added to the dimer subspace.

The dimer subspace then consists of all unique combinations of monomer determinants from the monomer subspaces, plus any sampled charge-transfer determinants. This systematic inclusion (or completion) guarantees that all necessary product determinants are present, satisfying strict size consistency.

3. Quantum Algorithm and Implementation Steps

The sc-HSB-QSCI approach is realized by integrating quantum hardware sampling with classical CI diagonalization. The essential workflow is as follows:

• State Preparation and Evolution: Begin with the base state (Hartree–Fock), evolve under $H$ for $k\Delta t$, measure, and record bitstrings.
• Subspace Assembly: Intra-monomer bitstrings are split and included in monomer subspaces; the dimer subspace is built from all pairs of monomer excitations, plus charge-transfer determinants.
• Symmetry Completion: Prior to subspace diagonalization, ensure that determinants in each subspace form a complete basis for the required $S_z$ and $S^2$ symmetry sectors.
• CI Diagonalization: Perform classical Hamiltonian diagonalization in the constructed subspaces to yield ground-state energies for dimer and monomers.
• Interaction Energy Calculation: Use the supramolecular formula (or, optionally, the dimer approach with well-separated fragments as reference) to extract interaction energies.

The implementation uses standard quantum chemistry and quantum computing frameworks (e.g., OpenFermion, Cirq), with fermion–qubit mapping via the Jordan–Wigner transform and sampling controlled by the number of Trotter steps and measurement shots.

4. Validation and Numerical Results

The sc-HSB-QSCI method was validated numerically on a spectrum of molecular systems:

• 4H/8H Clusters: For noninteracting clusters, the method yields $E_{\text{int}} \approx 0$ within $10^{-7}$ Hartree, confirming numerical size consistency.
• FH Dimer and FH–H$_2$O Complex: Hydrogen-bonded clusters were analyzed using active spaces for each fragment. Interaction energies computed with sc-HSB-QSCI converge rapidly to CAS-CI (complete active space configuration interaction) benchmarks as the number of sampled determinants grows, with errors below 0.04 kcal/mol for $K \geq 10$. Without the size consistency protocol, errors are an order of magnitude higher, even for large subspace sizes.
• Subspace Scaling: The method exploits the combinatorial structure of noninteracting monomers such that the dimer subspace size is the direct product of monomer subspaces. This scaling is crucial for large systems: for example, in 8H clusters, the augmented subspace remains efficient compared to the full configuration space.

A symmetry-completed subspace contributes to chemical accuracy and avoids discontinuities in potential energy surfaces across geometries.

5. Mathematical Formulation and Key Expressions

Table: Key Formulas in sc-HSB-QSCI

Quantity	Formula	Context
Interaction	$$E_\text{int} = E_\text{AB} - E_\text{A} - E_\text{B}$$	Supramolecular approach
Dimer Approach	$E_\text{int} = E_\text{AB} - E_{A \cdots B}$	Dimer at large separation as reference
Time Evolution	$U = e^{-iH\Delta t}$	Real-time Hamiltonian simulation
Dimer Wavefunction	$$|\Psi_\text{dimer}\rangle = \sum_{jk} c_j c_k\,|\psi_j\rangle\otimes|\psi_k\rangle$$	Product of monomer wavefunctions
Determinant Sampling	$$P(\text{sampled } |\psi_j\rangle \otimes |\psi_k\rangle) = |c_j|^2 |c_k|^2$$	Sampling from product state

For ideal (noninteracting) monomers, occurrence probabilities for product determinants decay as $|c_j|^2|c_k|^2$, so rare basis states might require $1/|c_j|^4$ measurements to appear, motivating deterministic completion.

6. Applications and Broader Impact

The size-consistent HSB-QSCI method has immediate application to benchmarking supramolecular interaction energies in hydrogen-bonded systems. Because it produces a systematic and geometry-consistent determinant basis, it is also suitable for potential energy surface calculations and for extending selected CI approaches (such as heat-bath CI) with a rigorous size-consistency guarantee.

In broader terms, this framework is compatible with fragment-based and multilevel methods (e.g., fragment molecular orbital, ONIOM), enabling quantum chemical treatment of systems beyond the reach of brute-force CI. By basing the determinant selection on real-time quantum evolution, it aligns naturally with near-term quantum hardware constraints, avoiding the need for fragile and resource-intensive variational state preparation.

7. Limitations and Future Directions

While sc-HSB-QSCI achieves size consistency and chemical accuracy for noninteracting and weakly interacting clusters, practical limitations arise for extremely large systems due to the growth of the combinatorial subspace. Mitigating factors include truncation strategies, targeted sampling, and integration with efficient classical postprocessing (e.g., heat-bath criteria within the symmetry-augmented subspace).

Future research may focus on:

• Extending the framework to treat multi-fragment systems and periodic supramolecular assemblies.
• Developing hybrid variants with other selected CI methods, as suggested by numerical success with "sc-HCI."
• Applying the approach to fragment-based properties, such as charge-transfer energies and multi-center interactions, where strict size consistency is critical.

The sc-HSB-QSCI method brings configuration interaction calculations into alignment with the exacting requirements of supramolecular quantum chemistry, providing a path for accurate, scalable quantum-enabled simulations of large molecular aggregates and complexes (Sugisaki, 27 Oct 2025).