Virtual Orbital Fragmentation (FVO)

• Virtual Orbital Fragmentation (FVO) is a technique that partitions the virtual orbital space into chemically intuitive fragments to facilitate efficient quantum simulations.
• It applies a many-body expansion to accurately recombine fragment energies, achieving chemical accuracy while significantly reducing the number of qubits required.
• FVO leverages parallel computation and screening protocols to deliver resource savings up to 66%, making large molecule and cluster simulations feasible on current NISQ hardware.

Virtual Orbital Fragmentation (FVO) is a systematic method designed to reduce quantum computational resources in electronic structure calculations by partitioning the virtual orbital space of a molecular system. Through a chemically-intuitive fragmentation of virtual orbitals and application of many-body expansion (MBE) techniques, FVO enables accurate recovery of correlation energy with substantially fewer qubits—a critical advance for quantum chemistry on Noisy Intermediate-Scale Quantum (NISQ) devices. FVO stands as a complementary strategy to spatial fragmentation methods, extending scalability and feasibility to larger molecules and clusters on currently available quantum hardware (Zahariev et al., 23 Oct 2025), while building on many-body expansion strategies in classical full configuration interaction (FCI) calculations (Eriksen et al., 2017).

1. Conceptual Framework and Mathematical Formulation

The essential objective of FVO is to minimize the quantum resource burden by partitioning the virtual (unoccupied) orbital space $V$ of a molecule into a set of non-overlapping fragments $V = V_1 \cup V_2 \cup ... \cup V_N$. Each fragment $V_i$ corresponds to a chemically meaningful subset of virtual orbitals, typically generated by localizing the virtual space using procedures such as Boys or Pipek-Mezey localization. Importantly, all fragment calculations retain the complete set of occupied orbitals $O$, thereby preserving the fidelity of chemical bonding and electron count.

To reconstruct the correlation energy, FVO employs an MBE over the virtual fragments. The general $n$-body expansion is written as:

$$E_{\mathrm{FVO}}^{(n)} = \sum_{i=1}^N \Delta E_i + \sum_{i<j}^N \Delta E_{ij} + \sum_{i<j<k}^N \Delta E_{ijk} + \dots$$

where the increments are:

• $\Delta E_i = E(O + V_i) - E(O)$
• $$\Delta E_{ij} = E(O + V_i + V_j) - E(O + V_i) - E(O + V_j) + E(O)$$
• $\Delta E_{ijk}$ involves further corrections for three-fragment interactions.

Each term $E(O + V_{\mathcal{S}})$ denotes a correlation energy computed for the total occupied space $O$ and the selected virtual subset(s) $\mathcal{S}$. The expansion avoids double-counting and rapidly converges to the full active-space energy with increasing number of bodies in the expansion.

2. Implementation Strategies and Screening Protocols

Efficient implementation of FVO depends on judicious fragment selection and screening of negligible energy increments. The construction of virtual orbital fragments is typically guided by chemical intuition and localization metrics; those orbitals associated with the same atomic region or functional group compose a natural fragment.

The many-body expansion is computationally demanding, with combinatorial growth in the number of fragment tuples at higher orders. To manage this, FVO leverages the locality of correlation—most higher-order terms contribute marginally. In practical approaches, a screening threshold $T_k$ is imposed at each expansion order $k$, following $T_k = T_{\text{init}} \cdot a^{k-1}$, with $a \geq 1$ and a small $T_{\text{init}}$, to systematically exclude tuples where $|\Delta E_\lambda| < T_k$. Screening ensures computational resources are focused on significant contributions, analogous to criteria in virtual orbital many-body expansions for classical FCI (Eriksen et al., 2017).

Parallelization is inherent to FVO: all fragment and tuple calculations are independent, enabling "embarrassingly parallel" execution. Each fragment or tuple energy can be computed concurrently, markedly reducing wall-clock time on multi-processor systems or quantum devices.

3. Integration with Quantum Algorithms and Hybrid Schemes

FVO is inherently suited to quantum-classical hybrid algorithms, especially the Variational Quantum Eigensolver (VQE). For each fragment or tuple, $E(O + V_i)$ or higher interactions are computed using VQE or other quantum eigensolver protocols, with the number of qubits scaling with the number of included orbitals.

For large molecular systems or clusters, FVO can be hierarchical. The Effective Fragment Molecular Orbital (EFMO) technique partitions the molecular system in real (spatial) space, and FVO is then applied within each spatial fragment (“Q-EFMO-FVO” scheme). This nested fragmentation yields multiplicative reductions in resource requirements, as the reduction from FVO compounds with the reduction from spatial decomposition.

The typical workflow involves:

1. Localizing and fragmenting the virtual space.
2. Computing 1-body, 2-body, and optionally higher-body fragment energies via quantum algorithms.
3. Applying MBE equations to sum contributions and reconstruct the total correlation energy.
4. For clusters, applying EFMO first, then FVO within each EFMO fragment.

4. Performance, Accuracy, and Resource Savings

Empirical benchmarks on small to medium molecular systems (e.g., water dimer, methanol, methylamine, ammonia) demonstrate that the 2-body FVO expansion achieves energy errors below 3 kcal/mol (96–99.5% correlation energy recovery), and 3-body expansions attain sub-kcal/mol accuracy. This level of precision is conventionally termed “chemical accuracy” and is sufficient for many applications in computational chemistry (Zahariev et al., 23 Oct 2025).

Qubit reduction is substantial: depending on the system, FVO reduces qubit requirements by 40–66% relative to using the full virtual space. For instance, systems that would conventionally require 96–128 qubits can, under FVO, be simulated with 48–74 qubits—well within reach for current 50–100 qubit NISQ hardware. Reductions in circuit depth (up to 62% in certain VQE simulations) further enhance the practicality of this approach on noisy devices.

The following table summarizes key performance outcomes as reported for selected systems:

Expansion Order	Mean Energy Error (kcal/mol)	Qubit Reduction
2-body	<3	40–66%
3-body	<1	40–66%

5. Relationship to Virtual Orbital Many-Body Expansions in Classical Electronic Structure Theory

FVO adapts foundational principles from virtual orbital many-body expansions, originally used to approach the FCI limit in classical electronic structure calculations (Eriksen et al., 2017). In the classical paradigm, many-body expansions are constructed over virtual orbital tuples, with each increment computed via small complete active space configuration interaction (CAS–CI) calculations. The resulting high-order MBE rapidly converges to the FCI energy, while screening and parallelism make computations tractable for systems with virtual spaces of several hundred orbitals.

FVO inherits the MBE formalism, partitioning energy increments over fragments of the virtual space and leveraging chemical locality for convergence. The transition to quantum computing context repurposes these strategies for minimization of qubit and circuit resources rather than classical memory and CPU time.

6. Limitations, Applicability, and Prospects

FVO is most effective for systems where correlation is predominately dynamic and local; for strongly multiconfigurational or highly delocalized systems, the convergence of the fragment-based MBE may be slower or require higher-body terms. The combinatorial scaling in fragment tuple evaluations, while mitigated by screening, remains a potential practical bottleneck at large system sizes or high fragment counts.

The approach does not truncate the occupied space—electron counting and valence structure are preserved—eliminating artifacts associated with bond-cutting in spatial fragmentations. FVO is particularly suited for current NISQ hardware constraints and can be systematically improved as hardware evolves by increasing expansion order or refining fragment definitions.

When combined with EFMO or similar spatial decomposition frameworks, FVO establishes a dual-axis fragmentation methodology that is applicable to large clusters, biomolecules, and condensed phase problems as NISQ-era quantum resources advance.

7. Significance for Quantum Chemical Computation

FVO provides a rigorous, systematically improvable, and parallelizable framework for reducing quantum resource requirements in ab initio calculations. Its chemical accuracy at low expansion order, substantial qubit and circuit savings, and compatibility with both hybrid quantum–classical and cluster-based decomposition techniques constitute a practical pathway towards scalable quantum chemistry on near-term devices. By enabling simulation of systems previously inaccessible to full active-space quantum algorithms, FVO is positioned as a central component in the ongoing development of quantum computational chemistry (Zahariev et al., 23 Oct 2025, Eriksen et al., 2017).