HI-VQE: Hybrid Quantum-Classical Eigensolver

• HI-VQE is a hybrid quantum-classical algorithm that iteratively refines electronic structure estimates by combining quantum sampling of subspaces with classical exact diagonalization.
• It employs a handover loop where quantum-prepared ansätze yield bitstring samples that are filtered and processed classically, thus avoiding direct Hamiltonian expectation measurements.
• Benchmark studies show chemically accurate energies with significantly reduced determinant spaces, enabling scalable simulations of complex molecular systems.

The Handover-Iterative Variational Quantum Eigensolver (HI-VQE) encompasses a class of hybrid quantum-classical algorithms for efficient, scalable, and noise-resilient solutions to electronic structure problems in quantum chemistry. HI-VQE leverages noisy intermediate-scale quantum (NISQ) hardware to sample subspaces of Slater determinants likely to contribute significantly to ground-state wavefunctions, and combines this with classical postprocessing—principally, exact diagonalization within these sampled subspaces—to estimate energies and update variational parameters. HI-VQE methods circumvent the measurement overhead and noise sensitivity characteristic of standard variational quantum eigensolvers (VQE) by never directly evaluating expectation values of Hamiltonians on quantum devices; instead, energy estimation and amplitude assignment are delegated to classical resources (Pellow-Jarman et al., 8 Mar 2025, Yoo et al., 11 Jan 2026, Ghasemi et al., 15 Jan 2026).

1. Hybrid Quantum-Classical Workflow

HI-VQE operates via an iterative handover loop between quantum and classical processors. At each iteration, the QPU prepares a parametrized ansatz $U(\theta)$, applies it to a reference (typically Hartree–Fock), and samples computational-basis bitstrings corresponding to electron configurations. The CPU subsequently filters these samples—removing or correcting those violating symmetry, electron number, or spin constraints—and defines or augments a compact subspace $C'$. The many-body Hamiltonian $H$ is projected into $C'$ to construct a reduced-dimension Hamiltonian $H_{C'}$, which is then exactly diagonalized. The resulting amplitudes $\{\omega_i\}$ and subspace energy $E$ drive determinant pruning, excitation generation (for subspace expansion), and parameter updates via a classical optimizer.

This loop avoids QPU-based Pauli-term measurements: energy is always evaluated classically, ensuring the estimate is noise-free up to diagonalization precision and floating-point artifacts. Barren plateau effects and measurement-induced noise, endemic to conventional VQE, are thus mitigated (Pellow-Jarman et al., 8 Mar 2025).

2. Theoretical and Algorithmic Foundations

At the core, HI-VQE seeks to minimize the ground-state energy of an electronic Hamiltonian of the form (second-quantized): $$H = \sum_{p,q} h_{pq} a_p^\dagger a_q + \frac{1}{2} \sum_{p,q,r,s} h_{pqrs} a_p^\dagger a_q^\dagger a_r a_s$$ The variational ansatz is implemented as $$|\psi(\theta)\rangle = U(\theta)|\psi_\mathrm{HF}\rangle$$, with energy objective $$E(\theta) = \langle\psi(\theta)| H |\psi(\theta)\rangle$$. In contrast to VQE, $E(\theta)$ is not measured on quantum hardware; instead, bitstring samples define a determinant subspace $\mathcal{C}$, in which $H_\mathrm{eff}$ is constructed: $$(H_\mathrm{eff})_{ij} = \langle V_i | H | V_j \rangle,\quad |V_i\rangle, |V_j\rangle \in \mathcal{C}$$ Exact diagonalization yields eigenvalues/eigenvectors providing both the energy $E$ and the expansion coefficients $\{w_i\}$ (Ghasemi et al., 15 Jan 2026). Pruning via an amplitude cutoff $\tau$ maintains only the most relevant determinants. Subspace expansion proceeds via generating (and screening) determinants from single/double excitations of leading contributors (Pellow-Jarman et al., 8 Mar 2025, Yoo et al., 11 Jan 2026).

A distilled pseudocode summary:

1. Prepare $|\psi(\theta)\rangle$ on QPU.
2. Sample $N$ raw bitstrings.
3. Classically filter for physicality (particle number, spin).
4. Form $\mathcal{C}$ from unique valid samples and extant core subspace.
5. Classically build and diagonalize $H_\mathrm{eff}$.
6. Prune negligible determinants and expand the subspace.
7. Update $\theta$ via classical optimization.
8. Repeat until convergence criteria (e.g., $|\Delta E| < \varepsilon$) are met (Pellow-Jarman et al., 8 Mar 2025, Ghasemi et al., 15 Jan 2026, Yoo et al., 11 Jan 2026).

3. Noise Mitigation and NISQ Compatibility

HI-VQE’s structure renders it uniquely resistant to noise sources that frustrate standard VQE, including shot and amplitude noise in Pauli expectation measurement. As only computational-basis measurements are required, even deep quantum circuits can be effectively exploited with limited susceptibility to readout or gate noise. In all workflow variants, invalid or noisy samples are repaired or discarded using configuration-recovery procedures, ensuring that only physically valid determinants populate the subspace (Pellow-Jarman et al., 8 Mar 2025, Yoo et al., 11 Jan 2026).

Empirical results demonstrate consistent noise robustness: HI-VQE achieves chemically accurate energies, with errors under $1$ mHa when benchmarked against state-of-the-art classical approaches and under hardware noise levels that degrade conventional VQE (Ghasemi et al., 15 Jan 2026, Yoo et al., 11 Jan 2026).

4. Scalability, Implementation, and Performance

HI-VQE is systematically extensible to large active spaces, well beyond the reach of FCI or complete active-space CI (CASCI). Reported benchmarks include:

• NH$_3$ in a (15o,10e) active space (30 qubits): CASCI requires $9\times10^6$ determinants; HI-VQE achieves comparable accuracy with $2\times10^5$ determinants—$\sim2.2\%$ of CASCI (Pellow-Jarman et al., 8 Mar 2025).
• Dissociation curves for systems such as Li$_2$S (24 qubits) and N$_2$ (40–56 qubits) yield chemical accuracy across bond-lengths with determinant subspaces orders of magnitude smaller than SHCI or DMRG (Pellow-Jarman et al., 8 Mar 2025, Yoo et al., 11 Jan 2026).
• For pyridine–Li$^+$ (44 qubits, (24e,22o) active space), HI-VQE converges where all-classical approaches fail due to resource constraints (Ghasemi et al., 15 Jan 2026).

Table: Energies and Determinant Counts for NH$_3$ (15o,10e, 30 qubits) (Pellow-Jarman et al., 8 Mar 2025) | Method | E_total (Ha) | N_dets | ΔE to CASCI (mHa) | |----------------|------------------|----------|-------------------| | RHF | −56.16003161 | 1 | +132.36 | | HiVQE $2\!\times\!10^{3}$ | −56.27829790 | 1,936 | +14.10 | | HiVQE $2\!\times\!10^{5}$ | −56.29215769 | 199,809 | +0.24 | | SHCI | −56.29239971 | 1,759,358| −0.12 | | CASCI | −56.29239989 | 9,018,009| 0.00 |

Convergence is typically reached in $5$–$40$ outer iterations, with $10^3$–$10^4$ QPU shots per iteration, and determinant subspaces compacted by up to three orders of magnitude relative to classical approaches while achieving errors below chemical thresholds (Pellow-Jarman et al., 8 Mar 2025, Ghasemi et al., 15 Jan 2026).

5. Extensions: Strong Correlation, Phase Estimation, and Cross-Instance Learning

Recent developments extend HI-VQE to strongly correlated systems (e.g., N$_2$, Fe–S clusters), yielding quantitative agreement with HCI and DMRG at substantially reduced computational cost. In N$_2$ dissociation (56 qubits; cc-pVDZ, (14e,28o)), HI-VQE matches HCI within $0.27$ mHa, and in [2Fe–2S] cluster simulations, chemical accuracy ($<1$ mHa) is reached within $6$–$9$ iterations, requiring $5\!\times\!10^7$ determinants—over an order of magnitude more compact than HCI reference (Yoo et al., 11 Jan 2026). The cost of classical diagonalization scales as $O(N_\mathrm{det}^3)$, but with $N_\mathrm{det}$ being orders smaller than in classical SCI or DMRG for similar accuracy.

In another variant, HI-VQE is used as a state-preparation strategy for iterative quantum phase estimation (IQPE). Here, a high-fidelity VQE-prepared state (high overlap with the true ground state) is "handed over" as input to IQPE, achieving exponential energy estimation precision with minimal sampling overhead. In the H$_4$-on-a-circle case, a minimally parametrized UCC ansatz (one parameter) yielded high ground-state overlap and enabled IQPE to recover exact energies with sub-millihartree errors (Halder et al., 2021).

An alternative handover paradigm, explored in the context of transfer learning across problem instances, involves the reuse and filtering of VQE parameter trajectories from a solved "seed" instance to accelerate convergence for related target instances. For unstructured graph-based MaxCut Hamiltonians, simple energy-based handover strategies yield performance statistically indistinguishable from random initialization, but more elaborate feature-based transfer schemes are suggested as a future direction (Hutchings et al., 2024).

6. Practical Implementation Considerations

Implementing HI-VQE entails:

• Active-space selection and orbital embedding, often via mean-field solutions (e.g., Hartree–Fock), using classical codes such as PySCF.
• Mapping fermionic Hamiltonians to qubit representations through Jordan–Wigner or Bravyi–Kitaev encodings, with Qiskit Nature or equivalent toolkits (Ghasemi et al., 15 Jan 2026).
• Generating shallow, hardware-efficient ansätze such as EfficientSU2 or excitation-preserving circuits, ensuring circuit depths compatible with the available quantum hardware.
• Ensuring basis-measurement strategy: only computational-basis (bitstring) measurements are required; no extensive Pauli grouping or tomography is necessary.
• Employing classical diagonalization routines (dense, sparse, or Lanczos), and classical optimizers (L-BFGS-B, gradient descent).
• Experience-driven tuning of subspace size control parameters and amplitude thresholds to balance accuracy and computational cost.

HI-VQE naturally incorporates error mitigation via postselection and filtering of samples, and additional NISQ-oriented error mitigation techniques (dynamical decoupling, readout error correction) may be used but are less critical than for canonical VQE (Pellow-Jarman et al., 8 Mar 2025, Ghasemi et al., 15 Jan 2026).

7. Outlook and Limitations

HI-VQE has established itself as a practical and accurate framework for quantum-accelerated electronic structure, particularly for multireference and strongly correlated systems. The interplay of quantum sampling and classical postprocessing enables advances along several key directions:

• Achieving chemical accuracy for systems with large active spaces and strong correlation, with quantum and classical resource requirements remaining tractable under current and near-term capabilities
• Avoiding core VQE bottlenecks—measurement noise, circuit-depth-induced barren plateaus, and exponential scaling of determinant spaces in FCI/SCI (Pellow-Jarman et al., 8 Mar 2025, Yoo et al., 11 Jan 2026, Ghasemi et al., 15 Jan 2026)
• Enabling extensions to larger systems via further development of diagonalization techniques, smarter subspace generation, and possible integration with tensor network solvers
• Use as a principled state preparation protocol for high-precision phase estimation algorithms (Halder et al., 2021)

Limitations include the eventual scaling bottleneck associated with classical diagonalization of very large subspaces ($N_\mathrm{det} \sim 10^8$), dependence of convergence rate on quantum sampling quality and amplitude cutoff scheduling, and the need for case-dependent parameter tuning for determinant pruning and subspace expansion. Research is ongoing to address these issues through hybridization with tensor network approaches, more adaptive handover and shot allocation protocols, and further error-robust ansatz engineering (Yoo et al., 11 Jan 2026, Ghasemi et al., 15 Jan 2026).