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arXiv:2610.40561

Grow, Compress, and Perturb: A Hybrid Quantum-Classical Pathway to Scalable Quantum Chemistry on Quantum Computers

Hugo D. Nogueira, Siwar Badreddine, Olivier Adjoua, Diata Traore, Guillaume Michel, Jean-Philip Piquemal

quant-phphysics.chem-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations• Revised Oct 30, 2026 (v2)
v1 · Oct 29, 2026v2 · Oct 30, 2026
arXivHTMLPDF

Abstract

While Variational Quantum Eigensolver (VQE) variants are among the leading candidates for electronic structure calculations on quantum computers, gradient-based operator selection limits their scalability. To solve this, we introduce the Hamiltonian-gradient-free VULCAN (Variational Upbuilding, Layered Compression, Adaptive N-body perturbation) layered VQE framework. Our method dynamically generates ansatz operators using Variational Double Bracket Flow (VDBF) equations. Combined with a linearly scaling minimal complete pool, it creates a pool 4-12 times smaller than QEB (Qubit-Excitation-Based) pools. By appending entire layers simultaneously, Hamiltonian gradient measurements are completely eliminated. Circuit depth is managed via a grow-then-compress scheme using efficient state-overlap measurements, and qubit requirements are reduced by projecting the Hamiltonian onto a dominant-determinant subspace. Using high-performance emulation up to 32 logical qubits, we achieve chemical accuracy at just 5-20% of the measurement cost of batched ADAPT-VQE for hydrogen chains, BeH2 and H2O. For the 32-qubit C3H6, where batched ADAPT-VQE does not reach chemical accuracy, a single VULCAN layer combined with a GPU-accelerated stochastic Epstein-Nesbet (ENPT2) correction does. This stochastic ENPT2 post-processing step enables early termination at chemical accuracy.

AI Summary

VULCAN addresses the measurement overhead of gradient-based operator selection in adaptive VQE for electronic structure. It constructs a Hamiltonian-informed Pauli pool using a truncated double-bracket-flow commutator, supplements it with a minimal complete pool, and appends and optimizes whole layers while freezing earlier parameters. Optional overlap-guided compression reduces circuit size; determinant-subspace projection reduces register size; classical stochastic Epstein–Nesbet perturbation theory (ENPT2) supplies an energy correction. Benchmarks use noiseless state-vector emulation of hydrogen chains, BeH$_2$, H$_2$O and a 32-qubit C$_3$H$_6$ active space. Without compression, chemical accuracy is reached using 5–20% of the cumulative Pauli-term evaluations of batched QEB-ADAPT-VQE, generally with deeper circuits. Compression lowers CNOT counts below that baseline in four of seven tested cases, all at stretched geometries, reaching 0.14 times the baseline for stretched H$_{10}$. Projected Hamiltonians reach chemical accuracy under exact diagonalization with 23–50% fewer qubits. Single-layer states plus stochastic ENPT2 reach chemical accuracy for 26-qubit H$_2$O and 32-qubit C$_3$H$_6$; the latter has a reported error of $1.0\times10^{-3}$ Ha. The resource estimates count Pauli-term evaluations, not shots; noisy optimization of layers containing roughly $10^3$ parameters remains untested. Only the minimal complete component scales linearly: added pool excitations scale as $O(N^4)$. Neither the pool nor projection preserves particle number or spin, and stretched-chain states can be contaminated. Classical projection preprocessing requires emulation, and ENPT2 retains exponential state-vector memory despite denominator sampling; its C$_3$H$_6$ denominator evaluations take 11.8 GPU-hours. Energies are referenced to finite-basis FCI, not the complete-basis limit.

Little sign of AI-generated textOverclaimsIncrementalMissing comparisonStrong assumptions

Central claim may not hold: Only the MCP component scales linearly; the full appended operator pool contains additional O(N^4) excitations that dominate its size. The abstract's pool-size reduction range of 4–12 disagrees with the C3H6 table entry, whose QEB/VULCAN ratio is approximately 24.63. The 5–20% measurement-cost result concerns uncompressed growth and Pauli-term evaluations, whereas compression can raise this proxy cost to 59% of the baseline and finite-shot costs are not established. The stochastic ENPT2 correction still requires full exponentially sized statevectors and classical ansatz emulation, so its demonstrated savings do not establish scalability beyond classical emulation. The stated pool generator vanishes identically for the particle-number-conserving electronic Hamiltonian because sum_i[Z_i,H] = -2[N_e,H] = 0, leaving the reported nonempty VDBF pool unexplained by the defined construction.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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While Variational Quantum Eigensolver (VQE) variants are among the leading candidates for electronic structure calculations on quantum computers, gradient-based operator selection limits their scalability. To solve this, we introduce the Hamiltonian-gradient-free VULCAN (Variational Upbuilding, Layered Compression, Adaptive N-body perturbation) layered VQE framework. Our method dynamically generates ansatz operators using Variational Double Bracket Flow (VDBF) equations. Combined with a linearly scaling minimal complete pool, it creates a pool 4-12 times smaller than QEB (Qubit-Excitation-Based) pools. By appending entire layers simultaneously, Hamiltonian gradient measurements are completely eliminated. Circuit depth is managed via a grow-then-compress scheme using efficient state-overlap measurements, and qubit requirements are reduced by projecting the Hamiltonian onto a dominant-determinant subspace. Using high-performance emulation up to 32 logical qubits, we achieve chemical accuracy at just 5-20% of the measurement cost of batched ADAPT-VQE for hydrogen chains, BeH2 and H2O. For the 32-qubit C3H6, where batched ADAPT-VQE does not reach chemical accuracy, a single VULCAN layer combined with a GPU-accelerated stochastic Epstein-Nesbet (ENPT2) correction does. This stochastic ENPT2 post-processing step enables early termination at chemical accuracy.
VULCAN addresses the measurement overhead of gradient-based operator selection in adaptive VQE for electronic structure. It constructs a Hamiltonian-informed Pauli pool using a truncated double-bracket-flow commutator, supplements it with a minimal complete pool, and appends and optimizes whole layers while freezing earlier parameters. Optional overlap-guided compression reduces circuit size; determinant-subspace projection reduces register size; classical stochastic Epstein–Nesbet perturbation theory (ENPT2) supplies an energy correction.
Benchmarks use noiseless state-vector emulation of hydrogen chains, BeH2_22​, H2_22​O and a 32-qubit C3_33​H6_66​ active space. Without compression, chemical accuracy is reached using 5–20% of the cumulative Pauli-term evaluations of batched QEB-ADAPT-VQE, generally with deeper circuits. Compression lowers CNOT counts below that baseline in four of seven tested cases, all at stretched geometries, reaching 0.14 times the baseline for stretched H10_{10}10​. Projected Hamiltonians reach chemical accuracy under exact diagonalization with 23–50% fewer qubits. Single-layer states plus stochastic ENPT2 reach chemical accuracy for 26-qubit H2_22​O and 32-qubit C3_33​H6_66​; the latter has a reported error of 1.0×10−31.0\times10^{-3}1.0×10−3 Ha.
The resource estimates count Pauli-term evaluations, not shots; noisy optimization of layers containing roughly 10310^3103 parameters remains untested. Only the minimal complete component scales linearly: added pool excitations scale as O(N4)O(N^4)O(N4). Neither the pool nor projection preserves particle number or spin, and stretched-chain states can be contaminated. Classical projection preprocessing requires emulation, and ENPT2 retains exponential state-vector memory despite denominator sampling; its C3_33​H6_66​ denominator evaluations take 11.8 GPU-hours. Energies are referenced to finite-basis FCI, not the complete-basis limit.
The Variational Quantum Eigensolver (VQE) is a leading algorithm for noisy intermediate-scale quantum (NISQ) devices, but its adaptive variants (e.g., ADAPT-VQE) suffer from severe classical simulation bottlenecks during the ansatz growth phase. Representing and exponentiating pool operators for multi-qubit systems constructs massive sparse matrices that quickly scale to millions of elements, choking classical memory bandwidth and CPU/GPU cycle capacity. In this work, we present a resource-efficient software-layer framework that completely bypasses dense matrix exponentiation by evaluating state updates through a deterministic fifth-order (O(5)) Taylor series expansion. This approach reframes the costly unitary evolution into a chained sequence of five lightweight, successive sparse matrix-vector multiplications scaling strictly as O(Nz), where Nz is the number of non-zero elements. We validate our framework using equilibrium BeH2 (14 qubits), equilibrium H2O (12 qubits), and strongly correlated asymmetrically stretched H2O molecular profiles under both Jordan-Wigner (JW) and Bravyi-Kitaev (BK) transformations. The simulation results demonstrate that the O(5) truncation maintains exceptional numerical fidelity, retaining a state fidelity > 0.999999 and matching absolute ground state energies down to subchemical accuracy, while effortlessly navigating matrix spaces exceeding 268 million structural elements. This framework provides a scalable, high-performance pathway for executing deep variational simulations on hardware platforms with constrained computational budgets.
The variational quantum eigensolver (VQE) algorithm can simulate the chemical systems such as molecules in the noisy-intermediate-scale quantum devices and shows promising applications in quantum chemistry simulations. The accuracy and computational cost of the VQE simulations are determined by the underlying ansatz. Therefore, the most important issue is to generate a compact and accurate ansatz, which requires a shallower parametric quantum circuit and can achieve an acceptable accuracy. The newly developed adaptive algorithms (AAs) such as the adaptive derivative-assembled pseudo-Trotter VQE (ADAPT-VQE) can solve this issue via generating compact and accurate ansatzes. However, these AAs show very low computational efficiency because they require a large number of additional measurements. Here we propose an amplitude reordering (AR) strategy to accelerate the promising but expensive AAs by adding operators in a “batched” fashion in a way that their order is still quasi-optimal. We first introduce the AR method into ADAPT-VQE and build the AR-ADAPT-VQE algorithm. We then endow the energy-sorting VQE (ES-VQE) algorithm with the adaptive feature and introduce the AR into AES-VQE to form the AR-AES-VQE algorithm. To demonstrate the performance of these algorithms, we calculate the dissociation curves of three small molecules, LiH, linear BeH 2, and linear H 6, by using (AR-)ADAPT-VQE and (AR-)AES-VQE algorithms. It is found that all of the AR-equipped AAs (AR-AAs) can significantly reduce the number of iterations and subsequently accelerate the calculations with a speedup of up to more than ten times without the obvious loss of accuracy. The final ansatz generated by the AR-AAs not only avoids extra circuit depth but also maintains the computational accuracy; sometimes the AR-AAs even outperforms their original counterparts.
Abstract Variational Quantum Eigensolver (VQE) methods aim to maximize the resources of existing noisy devices. However, they encounter difficulties in simulating molecules of industrially-relevant sizes, such as constructing the efficient ansatz. Adaptive variational algorithms (ADAPT-VQE) can solve this problem but with a significant increase in the number of measurements. Here, we reduce the measurement overhead of ADAPT-VQE via adding operator batches to the ansatz while keeping it compact. We reformulate the previously proposed qubit pool completeness criteria for the tapered qubit space and propose an automated pool construction procedure. Our numerical results indicate that reducing the qubit pool size from polynomial to linear conversely increases the number of measurements. We simulate a set of molecules, participating in the carbon monoxide oxidation processes using the statevector simulator and compare the results with VQE-UCCSD and classical methods. Our results pave the way towards usage of variational approaches for solving practically relevant chemical problems.
Simulating strongly correlated molecular systems on near-term quantum hardware remains challenging because current hardware offers limited quantum volume and moderate-fidelity qubits. One way to address this challenge is through bootstrap embedding (BE). Bootstrap embedding partitions molecules into smaller fragments that are then embedded in the ``bath'' of other fragments iteratively. Bootstrap embedding is appealing for quantum simulation because fragmentation reduces the qubit requirements of each fragment problem. In this work, we develop a quantum bootstrap embedding (QBE) workflow that uses variational quantum eigensolver (VQE) fragment solvers and examines the algorithmic choices that determine the success of the overall VQE-QBE procedure. To improve efficiency, we introduce FastAdaptVQE, a sparse-matrix-accelerated form of the adaptive variational quantum eigensolver (ADAPT-VQE) that replaces symbolic commutator evaluation with direct statevector calculations, and MatrixFreeAdaptVQE, a matrix-free extension that removes the sparse-matrix memory bottleneck that arises for larger fragments. We also explore modifying the ADAPT-VQE operator-selection step by replacing the purely greedy choice with a lookahead strategy. Benchmarks on H4_44​, F2_22​, and selected geometries of H6_66​ reach chemical accuracy, within 1 kcal/mol of BE results obtained with a full configuration interaction (FCI) solver. For nnn-butane, density-matching residuals converged, but energy errors remained outside chemical accuracy. These results show that combining QBE with VQE can accurately compute molecular energies, while density convergence alone does not ensure energy accuracy. This work lays the foundation for extending these energy calculations to larger molecular systems and quantum materials on near-term quantum hardware.
Circuit depth and measurement count are the main limitations for chemical applications of the leading variational quantum algorithms on quantum computers. We show that both resource requirements can be reduced by augmenting adaptive variational quantum algorithms with post-optimization corrective techniques. We focus on two such techniques---Hamiltonian-moment Krylov subspace diagonalization and the identity-augmented quantum self-consistent equation of motion (q-sc-EOM)---applied to optimized ADAPT-VQE states without further ansatz growth or nonlinear optimization. Across molecular systems studied with up to 20 qubits, first-order Krylov correction reduces the optimized iterations (and the corresponding number of adaptive layers) needed to reach the 1.6 mEh1.6~\mathrm{m}E_h1.6 mEh​ threshold by up to approximately sixfold, with a general trend toward larger reductions at higher determinant Rényi entropy. We also find measurement-cost reductions of up to approximately fortyfold using a suitable resource-cost model. Identity-augmented q-sc-EOM improves molecular potential-energy curves obtained from truncated ADAPT-VQE ansätze with limited expressivity and recovers accuracy from states produced by sampled operator ranking without additional gradient-based operator selection, providing an advantage when small pool gradients make selection measurement intensive. Together, these post-optimization corrections offer routes to improve energy accuracy and reduce adaptive circuit growth in quantum algorithms as quantum hardware advances.
We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.