Hierarchical Q-EFMO-FVO Quantum Approach

• Hierarchical Q-EFMO-FVO is a fragmentation method that integrates EFMO spatial partitioning with FVO orbital fragmentation to substantially reduce qubit requirements.
• The approach leverages the variational quantum eigensolver (VQE) on discrete fragments, achieving up to 66% qubit reduction while recovering nearly full correlation energy.
• Its scalable energy reconstruction and many-body expansion ensure chemically accurate simulations of molecular clusters on NISQ devices.

The Hierarchical Q-EFMO-FVO approach is a systematically layered fragmentation framework for quantum chemical calculations on noisy intermediate-scale quantum (NISQ) computers. By combining Effective Fragment Molecular Orbital (EFMO) spatial partitioning of molecular systems with Fragmentation of Virtual Orbitals (FVO) in orbital space, and deploying these within the Variational Quantum Eigensolver (VQE) paradigm, the approach achieves substantial qubit reduction—up to 66%—while maintaining near-complete recovery of correlation energy relative to full quantum computations. This hierarchical methodology offers a practical and scalable route to chemically accurate quantum simulation of molecules and molecular clusters on existing quantum hardware, addressing a principal bottleneck in quantum computational chemistry.

1. Virtual Orbital Fragmentation (FVO): Partitioning and Many-Body Expansion

FVO targets the exponential growth of the Hilbert space in quantum chemical simulations by fragmenting the typically much larger virtual orbital space into mutually disjoint, chemically intuitive subsets. Given the complete virtual space $V$, it is partitioned as $V = V_1 \cup V_2 \cup \ldots \cup V_N$, with $V_i \cap V_j = \varnothing$ for $i \neq j$. Each quantum calculation per fragment uses the complete occupied space $O$ together with the selected virtual subset $V_i$, preserving the integrity of chemical bonding while reducing simulation size.

The FVO strategy reconstructs the total correlation energy via a many-body expansion analogous to spatial fragmentation methods:

$$E_{\mathrm{FVO}}^{(n)} = \sum_i \Delta E_i + \sum_{i<j} \Delta E_{ij} + \sum_{i<j<k} \Delta E_{ijk} + \cdots$$

with

$\Delta E_i = E(O \cup V_i) - E(O)$

$$\Delta E_{ij} = E(O \cup V_i \cup V_j) - E(O \cup V_i) - E(O \cup V_j) + E(O)$$

This expansion ensures systematic convergence to the full correlation energy as higher-body terms are included and guarantees that all contributions are counted without redundancy.

2. Integration with the Variational Quantum Eigensolver

In the hierarchical Q-EFMO-FVO workflow, each FVO fragment corresponds to a separate quantum computation, typically simulated with the variational quantum eigensolver (VQE) and the unitary coupled-cluster ansatz. Each VQE problem is restricted to the full set of occupied orbitals and the selected fragment(s) from the virtual space, leading to reduced qubit requirements and lower circuit depths.

Qubit reductions in the 40–66% range compared to full simulations are reported, enabling otherwise inaccessible calculations on quantum processors with 50–100 qubits. Each fragment calculation can be independently distributed, supporting parallel computing architectures and improving overall workflow efficiency. The main challenge is rigorous error control when summing the fragment energies, especially with a variational algorithm, requiring judicious selection of chemically meaningful virtual fragments.

3. Role of the Effective Fragment Molecular Orbital (EFMO) Method

EFMO provides spatial fragmentation of molecular clusters, dividing a large system into spatial regions (e.g., monomers). It inherits from the Fragment Molecular Orbital (FMO) and Effective Fragment Potential (EFP) families, employing effective potentials to represent long-range interactions.

Within the Q-EFMO-FVO approach, EFMO decomposes multi-molecular systems into spatial fragments, and within each spatial fragment, FVO is applied to further partition the virtual orbitals. The two-tier structure multiplies computational savings: spatial fragmentation reduces the overall system size; orbital fragmentation shrinks the individual quantum problems, accommodating them on current quantum hardware. This complementarity is essential for tackling both the number of calculations and their individual quantum resource requirements.

4. Hierarchical Architecture and Energy Reconstruction

The hierarchical Q-EFMO-FVO method proceeds in a multi-scalar fashion:

1. Spatial decomposition of the entire system using EFMO into spatial fragments.
2. Orbital fragmentation within each spatial fragment, partitioning the virtual orbitials via FVO.
3. Independent VQE calculations on each fragment (full occupied + one or several FVO virtual subsets).
4. Reconstruction of the total electronic energy using the following scheme:

$$E_{\mathrm{total}} \approx E_{\mathrm{EFMO}} + E_{\mathrm{FVO}}$$

with

$$E_{\mathrm{FVO}} = \sum_i \Delta E_i + \sum_{i<j} \Delta E_{ij} + \sum_{i<j<k} \Delta E_{ijk} + \cdots$$

This layered process ensures systematic inclusion of all significant correlation effects, with high accuracy even when only the 2-body or 3-body FVO terms are included.

5. Benchmark Applications and Quantitative Performance

The method has been validated on molecular species such as acetaldehyde, water dimer, methylamine, methanol, hydrogen peroxide, and ammonia, covering diverse bonding scenarios and basis set sizes. In these systems, virtual orbitals typically dominate the total orbital and thus qubit count (66–86%). Table-based results show FVO reduces qubit requirements to 48–74 for systems that would otherwise require 96–128.

Error analysis demonstrates:

• 2-body FVO expansion: Errors are always under 3 kcal/mol, reaching 96–99.5% recovery of full correlation energy.
• 3-body FVO expansion: Errors drop below 1–2 kcal/mol, with full (sub-kcal/mol) chemical accuracy.

These performance metrics are consistent across CCSD and CCSD(T) frameworks and are compatible with VQE-based quantum implementations.

6. Implications for Quantum Computing and Prospects

The hierarchical Q-EFMO-FVO method enables quantum simulations of molecular systems previously inaccessible to NISQ processors, primarily by reducing both the required number of qubits and the quantum circuit depth necessary for each fragment calculation. This is critical for practical NISQ chemistry, where device decoherence and qubit limitations are severe constraints.

Further avenues suggested include:

• Adaptive fragmentation based on importance metrics to optimize fragment selection.
• Integration with quantum error mitigation to address hardware noise.
• Extension to excited state computations (for spectroscopy, photochemistry).
• Hybridization with techniques for active space selection or frozen natural orbitals.
• Empirical demonstration on hardware as device capabilities mature.

This suggests that hierarchical Q-EFMO-FVO forms a foundational strategy for scalable and accurate quantum molecular simulation on NISQ platforms, with a pathway toward even more sophisticated applications as both algorithms and hardware advance.