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Quantum Science and Technology

Training-free operator selection for quantum circuit construction from a heterogeneous-graph representation

Li Hao, Chaoqun Ji, Mengbo Fu

Published Sep 28, 2026• 0 citations
Publisher

Abstract

Abstract Automated construction of variational quantum circuits remains a major bottleneck in quantum chemistry, largely because adaptive VQE methods repeatedly screen large operator pools using quantum-gradient criteria. For a canonical Hartree–Fock reference and normalized double-excitation generators, the initial ADAPT gradient magnitude is exactly the corresponding antisymmetrized two-electron Hamiltonian coupling. We encode this static signal as an excitation-edge feature of a heterogeneous graph, enabling a training-free classical ranking without current-state candidate-gradient measurements. Across statevector benchmarks from eight to twenty qubits, incremental construction reaches chemical accuracy with operator counts close to full gradient-based selection while avoiding repeated quantum screening. Supervised and residual models are used as controls to test whether learned corrections improve this low-cost rule. Static learned readouts provide no consistent held-out advantage, whereas state-conditioned features improve selected teacher-forced ranking metrics but require additional information and do not establish a closed-loop reduction in operator count. As a complementary higher-rank contribution, we derive an exact shared-residual formulation that reduces Hamiltonian applications for triple- and quadruple-excitation screening from 2 C or C + 1 to one while preserving all scores and rankings. Results on stretched H 6 and N 2 benchmarks show that these higher-rank operators can repair a singles-and-doubles expressivity limit.

DOI
10.1088/2058-9565/aead18

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Abstract Automated construction of variational quantum circuits remains a major bottleneck in quantum chemistry, largely because adaptive VQE methods repeatedly screen large operator pools using quantum-gradient criteria. For a canonical Hartree–Fock reference and normalized double-excitation generators, the initial ADAPT gradient magnitude is exactly the corresponding antisymmetrized two-electron Hamiltonian coupling. We encode this static signal as an excitation-edge feature of a heterogeneous graph, enabling a training-free classical ranking without current-state candidate-gradient measurements. Across statevector benchmarks from eight to twenty qubits, incremental construction reaches chemical accuracy with operator counts close to full gradient-based selection while avoiding repeated quantum screening. Supervised and residual models are used as controls to test whether learned corrections improve this low-cost rule. Static learned readouts provide no consistent held-out advantage, whereas state-conditioned features improve selected teacher-forced ranking metrics but require additional information and do not establish a closed-loop reduction in operator count. As a complementary higher-rank contribution, we derive an exact shared-residual formulation that reduces Hamiltonian applications for triple- and quadruple-excitation screening from 2 C or C + 1 to one while preserving all scores and rankings. Results on stretched H 6 and N 2 benchmarks show that these higher-rank operators can repair a singles-and-doubles expressivity limit.
Adaptive variational quantum algorithms like ADAPT-VQE construct tailored ansätze by iteratively selecting operators from a pool using gradient-based criteria. While this avoids oversized parameter spaces, repeatedly scanning the full pool incurs a classical cost that scales linearly with pool size-a major bottleneck for systems with long-range interactions or large operator sets. Here, we reformulate adaptive operator selection as a graph-based decision problem and introduce a graph neural network (GNN) policy that predicts the next entangling operator directly from the interaction graph and state-dependent observables. Training data are generated from exact simulations of disordered long-range spin chains, using gradient magnitudes as supervision signals. The learned policy accurately reproduces the dominant structure of the greedy gradient-based selection rule, significantly outperforming heuristics based solely on interaction strength. Integrated into a variational quantum eigensolver (VQE) workflow, this GNN-VQE approach achieves energy errors close to standard ADAPT-VQE while drastically reducing full-pool gradient evaluations. To test transferability beyond spin models, we evaluate the policy on small active-space molecular benchmarks (LiH and BeH_222). We find the GNN is highly effective as a shortlist generator: exact rescoring over just a few GNN-proposed candidates recovers near-oracle rollout behavior while searching only a small fraction of the pool. These results demonstrate that adaptive circuit construction contains learnable structure that can be exploited to accelerate variational quantum algorithms.
Designing compact and accurate circuits for the variational quantum eigensolver (VQE) is a central challenge in near-term quantum chemistry. Existing adaptive methods such as ADAPT-VQE design circuits by iteratively selecting operators from a predefined pool guided by gradient information and greedy heuristics. In this work, we adopt differentiable quantum architecture search (DQAS) as a circuit design framework based on the UCCSD operator pool, and introduce two complementary strategies: a global mode that simultaneously optimizes all operator selections, and a layerwise mode that constructs circuits incrementally while preserving previously learned structure. By relaxing discrete operator selection into a continuous differentiable optimization, DQAS enables gradient-based exploration over the combinatorial space of UCC circuit architectures. Benchmarks on BeH2, H4, LiH, H6, and H2O (8-14 qubits) show that both strategies achieve higher accuracy and fewer CNOT gates than ADAPT-VQE in the compact circuit regime, with up to 2.7-fold accuracy improvement for H2O and CNOT reductions of 13-17% at equivalent circuit depths. Benchmarks on the qubit-excitation-based (QEB) operator pool confirm that both advantages generalize beyond UCCSD. These results demonstrate that differentiable architecture search provides an effective and generalizable framework for designing accurate and compact VQE circuits in near-term quantum chemistry.
Predicting many molecular properties requires accurate ground-state energies of molecular Hamiltonians. Quantum-selected configuration interaction (QSCI) obtains such energies by measuring an input state to select important electronic configurations and classically diagonalizing the Hamiltonian in the subspace they span. ADAPT-QSCI, an adaptive extension of QSCI, improves the input state through iterative circuit growth. For single-reference molecules, the sampling distribution is dominated by the Hartree-Fock (HF) configuration, so most measurements reproduce a configuration already known. We reduce this redundancy by placing the HF configuration in the diagonalization subspace classically, before any measurement. This invalidates the operator-selection criterion of ADAPT-QSCI, and we derive a consistent replacement with energy and gradient in closed form, adding no query to the quantum device. At the per-iteration shot budgets previously used for ADAPT-QSCI, the proposed scheme reaches chemical accuracy with approximately 63%63\%63% and 43%43\%43% of the CNOT gates for H4\mathrm{H}_4H4​ and H6\mathrm{H}_6H6​, and 44%44\%44% and 40%40\%40% of the total shots. The advantage depends on the budget, however: in the infinite-shot limit the H4\mathrm{H}_4H4​ CNOT count increases to 125%125\%125% of that of ADAPT-QSCI. These results identify the per-iteration shot budget as a key design variable governing resource savings in QSCI-based algorithms.
Simulating strongly correlated fermionic pairing in the presence of rotational and pair-breaking fields requires deep quantum circuits. We present a graph-structured variational quantum eigensolver whose pair-transfer and single-excitation rotations follow the nonzero pairing and one-body mixing edges of the Hamiltonian. The circuit conserves particle number exactly and uses one parameter per retained edge. We benchmark one layer against exact fixed-number diagonalization for 1500, M=8M=8M=8 Hamiltonians represented by 16 qubits. Its mean energy error rises from 0.86 keV without one-body driving to 691 keV at the strongest drive; the one-layer circuit loses accuracy as pair breaking strengthens. In a matched eight-qubit comparison, the graph circuit reaches the 0.42 keV high-drive error of a pair-plus-all-singles circuit with 8 instead of 34 parameters. Fixed-number Adaptive Derivative-Assembled Pseudo-Trotter VQE reaches 0.056 keV with 304 decomposed controlled-NOT gates and iterative pool screening, while a 52-parameter, single-repetition unitary coupled-cluster singles-and-doubles circuit gives 17.6 keV with 2752 such gates. At a separate twelve-qubit point, a second graph layer reduces the high-drive error from 378 to 35 keV. Across the exact grid, an off-diagonal pair-coherence scale tracks the leading pair-density eigenvalue, condensate fraction, and interaction energy. Cranked zirconium Hamiltonians provide the benchmark instances and tune the strength of one-body pair breaking.
Training parameterized quantum circuits requires balancing optimization progress with limited evaluation budgets and the statistical uncertainty of near-term quantum hardware. We present a modular benchmarking framework that separates quantum search-direction estimation from classical parameter-update rules, allowing their interactions and sensitivities to be examined independently. We evaluate a suite of gradient-based, stochastic, and derivative-free optimizers across combinatorial optimization using the Quantum Approximate Optimization Algorithm (QAOA), supervised quantum machine learning using Iris classification and a quantum convolutional neural network (QCNN) for binary MNIST classification, and quantum chemistry using the variational quantum eigensolver (VQE) for molecular hydrogen. By separating objective evaluations from sample-circuit costs under finite-shot simulation and physical execution on a 156-qubit processor, we compare optimizer behavior across workloads with different parameter counts and measurement requirements. Four selected simulator and hardware case studies report terminal and best observed objectives separately, alongside post-training classifier accuracies. The comparisons include selected MaxCut and hydrogen trajectories and illustrate workload-dependent objective changes and evaluation costs, while the simulator benchmark compares both peak and mean performance across three seeds. The hardware runs do not establish an optimizer ranking or isolate a causal effect of device noise.
Determining the ground state of a many-body Hamiltonian is a central problem across physics, chemistry, and combinatorial optimization, yet it is often classically intractable due to the exponential growth of Hilbert space with system size. Even on fault-tolerant quantum computers, quantum algorithms with convergence guarantees -- such as quantum phase estimation and quantum subspace methods -- require an initial state with sufficiently large overlap with the true ground state to be effective. Variational quantum eigensolvers (VQEs) are natural candidates for preparing such states; however, standard VQEs typically exhibit poor generalization, requiring retraining for each Hamiltonian instance, and often suffer from barren plateaus, where gradients can vanish exponentially with circuit depth and system size. To address these limitations, we propose an end-to-end representation learning framework that combines a graph autoencoder with a classical neural network to generate VQE parameters that generalize across Hamiltonian instances. By encoding interaction topology and coupling structure, the proposed model produces high-overlap initial states without instance-specific optimization. Through extensive numerical experiments on families of one- and two-local Hamiltonians, we demonstrate improved generalization and trainability, manifested as reduced test error and a significantly milder decay of gradient variance. We further show that our method substantially accelerates convergence in quantum subspace-based eigensolvers, highlighting its practical impact for downstream quantum algorithms.