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Generative Quantum-inspired Kolmogorov-Arnold Eigensolver

Yu-Cheng Lin, Yu-Chao Hsu, I-Shan Tsai, Chun-Hua Lin, Kuo-Chung Peng, Jiun-Cheng Jiang, Yun-Yuan Wang, Tzung-Chi Huang, Tai-Yue Li, Kuan-Cheng Chen, Samuel Yen-Chi Chen, Nan-Yow Chen

arXiv:2605.04604v1quant-phcs.LG

TL;DR

Scalable quantum chemistry workflows need eigensolvers that manage the classical parameter and memory costs of coupled HPC-quantum processing. GQKAE replaces parameter-heavy generative-model transformations with compact quantum-inspired Kolmogorov-Arnold modules while preserving autoregressive operator selection and QSCI evaluation, achieving chemical accuracy with 66% lower trainable parameters and memory than the GPT-2 baseline.

  • Problem

    Scalable quantum chemistry eigensolvers must manage the parameter and memory footprint of classical components alongside quantum accuracy and circuit depth.

  • Method

    GQKAE integrates hybrid quantum-inspired Kolmogorov-Arnold modules into the GQE generative backbone while preserving autoregressive operator selection and QSCI evaluation.

  • Results

    GQKAE achieves chemical accuracy across molecular benchmarks while reducing trainable parameters and parameter memory by approximately 66% relative to the GPT-2 baseline and improving wall-time performance.

  • Takeaways & Limitations

    The QKAN-based backbone provides a scalable route for reducing classical-side memory and runtime overhead in HPC-quantum co-design.

Abstract

from arXiv · show

High-performance computing (HPC) is increasingly important for scalable quantum chemistry workflows that couple classical generative models, quantum circuit simulation, and selected configuration interaction postprocessing. We present the generative quantum-inspired Kolmogorov-Arnold eigensolver (GQKAE), a parameter-efficient extension of the generative quantum eigensolver (GQE) for quantum chemistry. GQKAE replaces the parameter-heavy feed-forward network components in GPT-style generative eigensolvers with hybrid quantum-inspired Kolmogorov-Arnold network modules, forming a compact HQKANsformer backbone. The method preserves autoregressive operator selection and the quantum-selected configuration interaction evaluation pipeline, while using single-qubit DatA Re-Uploading ActivatioN modules to provide expressive nonlinear mappings. Numerical benchmarks on H4, N2, LiH, C2H6, H2O, and the H2O dimer show that GQKAE achieves chemical accuracy comparable to the GPT-based GQE architecture, while reducing trainable parameters and memory by approximately 66% and improving wall-time performance. For strongly correlated systems such as N2 and LiH, GQKAE also improves convergence behavior and final energy errors. These results indicate that quantum-inspired Kolmogorov-Arnold networks can reduce classical-side overhead while preserving circuit-generation quality, offering a scalable route for HPC-quantum co-design on near-term quantum platforms.

I. INTRODUCTION · II. RELATED WORK · A. Classical and quantum methods for quantum chemistry

The paper motivates HPC-enabled quantum chemistry workflows and introduces GQKAE, which integrates compact HQKAN modules into GQE to reduce classical overhead while preserving quantum-chemistry performance. It situates this approach among classical electronic-structure methods, VQE variants, and generative quantum eigensolvers.

  • I. INTRODUCTION: HPC must efficiently couple electronic-structure preprocessing, quantum-circuit simulation, model training, and selected configuration-interaction postprocessing for scalable quantum chemistry.These costs become especially important for near-term algorithms as active spaces, operator pools, and generated sequences grow.
  • I. INTRODUCTION: Ground-state estimation is central to quantum chemistry, while VQE-based approaches face limitations from ansatz design, expressivity, noise, and barren plateaus.VQE performance depends strongly on continuous-parameter optimization and circuit ansatz choices.
  • I. INTRODUCTION: GQE reformulates ground-state preparation as discrete circuit generation, with a classical model learning unitary sequences whose sampled circuits guide training through their energies.Its trainable parameters reside in the classical generator rather than directly in the quantum circuit.
  • I. INTRODUCTION: GPT-style GQE can become a scalability bottleneck because dense position-wise nonlinear transformations increase parameter count, memory use, and training cost.The overhead grows with operator-vocabulary size and generated-sequence length, affecting HPC-enabled workflows.
  • II. RELATED WORK: QKAN uses learnable edge functions implemented by DARUAN modules, while HQKAN inserts a QKAN latent processor between classical encoder and decoder components.The modules are inspired by single-qubit data re-uploading circuits whose Fourier spectrum expands with repeated encoding.
  • I. INTRODUCTION: GQKAE integrates HQKAN into GQE’s generative backbone as an HQKANsformer, preserving autoregressive operator selection and the QSCI evaluation pipeline.The method replaces parameter-heavy nonlinear transformations with compact QKAN-based modules and is applied to H4, N2, LiH, C2H6, H2O, and the H2O dimer.
  • I. INTRODUCTION: Approximately 66% reductions in trainable parameters and parameter memory, together with wall-time speedups, are reported while maintaining chemical accuracy and comparable quantum resource costs to GQE.The benchmarks span bond dissociation, conformational variation, and intermolecular interaction.
  • A. Classical and quantum methods for quantum chemistry: Classical quantum-chemistry methods range from efficient HF baselines and high-accuracy CCSD(T) for weak correlation to strongly correlated FCI, CASSCI, and selected-configuration approaches.VQE similarly progresses from fixed ansätze to adaptive constructions, while barren-plateau-limited optimization motivates shifting learning toward classical components.

B. Machine-learning-based quantum circuit design and KAN architectures · III. PRELIMINARIES · A. Generative Quantum Eigensolver

The section surveys machine-learning approaches to quantum circuit design, motivates quantum-inspired KAN architectures, and introduces GQE as autoregressive circuit generation over operator sequences. GQE evaluates generated circuits through their prepared states and Hamiltonian energies, with GPT-2 parameterizing the baseline conditional distributions.

  • B. Machine-learning-based quantum circuit design and KAN architectures: Machine-learning approaches to quantum circuit design comprise reinforcement-learning, differentiable, and one-shot quantum architecture search methods.RL constructs circuits gate-by-gate, whereas differentiable and one-shot methods enable gradient-based search through continuous relaxation or parameter sharing.
  • B. Machine-learning-based quantum circuit design and KAN architectures: KAN replaces fixed node activations with learnable univariate functions, providing expressive approximation with favorable parameter efficiency.KAN-based methods have been applied to molecular property prediction, VQE architecture search, and transformer models, though scalability remains limited by trainable-activation growth.
  • B. Machine-learning-based quantum circuit design and KAN architectures: The method adopts quantum-inspired KAN modules with DARUAN activations and HQKAN to retain expressive edge parameterization while supporting scalability.HQKAN is described as scalable to large language models.
  • A. Generative Quantum Eigensolver: GQE recasts quantum ground-state search as learning a probability distribution over circuits, unlike VQE’s continuous optimization within a predetermined circuit family.A trainable generative model sequentially selects operators from a predefined pool to construct candidate circuits.
  • A. Generative Quantum Eigensolver: GQE uses an operator pool derived from the UCCSD ansatz, with each candidate circuit represented by an ordered sequence of selected operator indices.For circuit length L, the sequence specifies an ordered composition drawn from the operator pool G.
  • A. Generative Quantum Eigensolver: The selected operators compose a circuit that acts on an initial reference state to prepare a trial state evaluated through the target Hamiltonian’s expectation value.This evaluation measures the quality of the sampled circuit.
  • A. Generative Quantum Eigensolver: GQE increases the probability of operator sequences associated with lower-energy states through autoregressive conditional modeling, using a GPT-2 architecture as baseline.Operators are generated token by token, conditioned on previously selected operators from the predefined pool.

B. Quantum-selected configuration interaction

QSCI evaluates generated trial circuits by sampling computational-basis measurement outcomes, constructing a truncated determinant subspace, and classically diagonalizing the projected Hamiltonian. Its resulting subspace energy supplies the reward signal, with lower energies receiving higher rewards.

  • Procedure: QSCI evaluates a generated trial state by constructing a truncated subspace from measurement outcomes and performing classical Hamiltonian diagonalization within it.This avoids directly estimating the full Hamiltonian expectation value.
  • Procedure: Repeated computational-basis measurements produce sampled bitstrings that define a determinant subset D(j) corresponding to Slater determinants.The sampled bitstrings are used to identify the determinants retained for QSCI.
  • Procedure: At most dmax determinants are retained according to sampling frequency, defining the truncated set Ddmax(j) and its spanned subspace S(j).The determinant cap controls computational cost.
  • Energy evaluation: Within S(j), QSCI constructs the projected Hamiltonian matrix and obtains the energy by solving its eigenvalue problem.The matrix elements are evaluated between determinants in Ddmax(j).
  • Optimization signal: The QSCI energy evaluates generated circuits and defines rewards for sampled operator sequences, assigning higher rewards to circuits with lower subspace energies.This reward links the generative model to QSCI evaluation in the GQE optimization loop.

C. Quantum-inspired Kolmogorov–Arnold Network

QKAN constructs multivariate mappings from learnable edge-wise univariate transformations implemented by single-qubit DARUAN modules. This design provides expressive nonlinear mappings with relatively few trainable parameters and shifts representation away from fixed node-wise activations.

  • QKAN construction: QKAN builds multivariate mappings from learnable univariate transformations associated with edges rather than fixed node-wise activation functions.Each edge function is associated with a learnable transformation between an input coordinate and an output unit.
  • DARUAN activation: Single-qubit DARUAN modules implement the edge activations through parameterized data re-uploading circuits and measurement observables.For scalar input x, the activation is defined as ϕ(x; ϑ) = ⟨0| U†(x; ϑ) M U(x; ϑ)|0⟩.
  • QKAN construction: Stacked QKAN layers aggregate edge-wise transformations and summations into a hierarchical nonlinear map.The layer mapping is written as x_l+1 = Φ_l(x_l), with Φ_l collecting transformations and summations across layer l.
  • DARUAN activation: Repeated data re-uploading induces a rich Fourier spectrum, enabling highly nonlinear mappings while maintaining parameter efficiency.DARUAN maps scalar inputs to learnable nonlinear responses with expressivity and relatively few trainable parameters.
  • Design comparison: Compared with standard multilayer perceptrons, QKAN replaces node-wise affine projections with adaptive edge-wise nonlinear operators suited to expressive, parameter-efficient conditional transformations.This design is presented as particularly suitable for generative sequence modeling.

D. HQKAN architecture · IV. METHOD · A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver

GQKAE integrates quantum-inspired Kolmogorov–Arnold modules into the GQE-for-QSCI workflow, replacing GPT-style feed-forward transformations while preserving autoregressive circuit generation and QSCI evaluation. Its encoder–QKAN–decoder design uses a bottleneck latent space and DARUAN activations to provide a parameter-efficient nonlinear mapping.

  • D. HQKAN architecture: HQKAN combines classical and quantum-inspired neural computation in an encoder–processor–decoder architecture with a compact latent representation.The latent representation is processed by a KAN-based module before the decoder reconstructs the output representation.
  • IV. METHOD: The GQE-for-QSCI workflow generates operator sequences, constructs candidate circuits, measures quantum states for Slater determinants, and classically diagonalizes the sampled subspace.GQKAE targets the generative backbone rather than modifying the QSCI post-processing pipeline.
  • A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver: GQKAE replaces the conventional FFN in each transformer block with an HQKAN module to reduce trainable parameters while preserving autoregressive sequence modeling.The standard GQE formulation remains unchanged at the level of circuit generation.
  • A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver: The standard GPT-2 FFN uses learnable matrices W1 and W2, and its intermediate dimension dff is typically much larger than d, making it a substantial contributor to total parameters.The FFN is described as a two-layer nonlinear mapping with activation σ(·).
  • A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver: HQKAN projects hidden states into a latent space with dz < d, applies QKAN transformations through edge-wise DARUAN activations, and projects the result back to the original hidden dimension.The latent dimension dz is defined by the encoder projection, while each transformed latent component is implemented by a DARUAN module.
  • A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver: The HQKAN replacement leaves the GPT-style autoregressive backbone structure intact while substituting its nonlinear feed-forward transformation with a latent-space QKAN processor.Layer normalization and self-attention operations are inherited from the GPT-style architecture.
  • A. Generative Quantum-inspired Kolmogorov–Arnold Eigensolver: O(log(1/ϵ)) complexity scaling for approximation error ϵ supports the encoder–QKAN–decoder HQKAN module as a parameter-efficient alternative to the standard GPT-2 feed-forward layer.GQKAE consequently extends GQE by modeling the generative distribution over circuit operators with an HQKANsformer backbone.

B. GQKAE for QSCI

GQKAE retains the QSCI evaluation pipeline while replacing only the generative policy with an HQKANsformer-based autoregressive generator. Training uses clipped GRPO, with sequence-level QSCI energy determining rewards and normalized advantages for token-level updates.

  • QSCI evaluation: The HQKANsformer generates operator sequences whose circuits produce trial states for truncated-determinant construction and QSCI evaluation.The framework retains the original QSCI evaluation procedure while modifying the generative policy used to produce candidate circuits.
  • Policy training: Clipped GRPO trains the autoregressive policy at the token level using the quality of each complete generated circuit as the learning signal.The objective uses a token-wise importance ratio between the current policy and the reference policy that generated the samples.
  • Reward construction: Rewards are defined as r(m) = −EQSCI(j(m)), so sequences with lower QSCI energies receive larger rewards.The reward is computed for each sampled operator sequence before within-batch standardization.
  • Reward construction: The normalized advantage is computed by subtracting the batch mean reward and dividing by its standard deviation.The mean and standard deviation are calculated across the rewards in the sampled batch.

V. NUMERICAL RESULTS · A. Computational Details and Experiment Setup

The numerical study evaluates GQKAE against the original GQE baseline across molecular geometries using GPU-accelerated simulations. The setup combines compressed QKAN-based policy modeling, fixed sampling and optimization protocols, QSCI post-processing, and classical and VQE reference methods.

  • A. Computational Details and Experiment Setup: GQKAE was evaluated against the original GQE baseline on an NVIDIA HGX H200 system.The benchmark covered multiple molecular systems and geometric properties.
  • A. Computational Details and Experiment Setup: The study scanned bond lengths, bond angles, and intermolecular distances for H4, N2, LiH, C2H6, H2O, and the H2O dimer.The corresponding configurations used 8, 16, 20, 12, 16, and 16 qubits, respectively.
  • A. Computational Details and Experiment Setup: The policy network used a decoder-only GPT-2 architecture with a QKAN layer embedded in a highly compressed latent space.The supplied passage describes this design as controlling quantum-inspired activation parameter overhead.
  • A. Computational Details and Experiment Setup: The input was projected to a logarithmic latent dimension of dlatent = 12 before QKAN nonlinear transformations.FlashQKAN and cuQuantum were used to accelerate QKAN tensor-contraction paths.
  • A. Computational Details and Experiment Setup: M = 10 circuits per iteration, Nshots = 105, and Niter = 100 defined the sampling and training schedule.Optimization used GRPO and AdamW with learning rate 5 × 10−6, weight decay of 0.01, 30 policy updates per batch, and a 1.2 repetition penalty; QSCI used dmax = 2000 with symmetry completion.
  • A. Computational Details and Experiment Setup: Classical references included restricted HF, CCSD, SCI, and CASCI energies computed through PySCF.CASCI served as the variational reference because the active-space FCI dimension did not exceed 5×103.
  • A. Computational Details and Experiment Setup: The quantum baseline was VQE with a UCCSD ansatz simulated via CUDA-Q.VQE used COBYLA with maximum iterations set to 5000, starting from a zero-initialized parameter state applied to the HF reference.

B. Convergence Dynamics of Circuit Generation · C. Results of Potential Energy Surfaces

GQKAE and GQE both reach chemical accuracy, while GQKAE shows lower final errors and faster early convergence in several strongly correlated systems. Across potential energy surfaces, both methods reproduce molecular energy profiles and wells closely, including multireference and weak-interaction regimes.

  • B. Convergence Dynamics of Circuit Generation: Both GQE and GQKAE ultimately reach the chemical accuracy threshold across the evaluated molecular systems.The convergence metric is the lowest energy error relative to exact CASCI energy achieved up to each iteration.
  • B. Convergence Dynamics of Circuit Generation: H4 produces nearly flat error curves, with both methods finding highly accurate states from the first iterations.The H4 optimization landscape is described as relatively trivial, and errors remain below the chemical-accuracy margin immediately.
  • B. Convergence Dynamics of Circuit Generation: GQKAE achieves lower final energy errors than GQE for N2, LiH, and H2O.The advantage appears as electronic-structure complexity and correlation strength increase.
  • B. Convergence Dynamics of Circuit Generation: GQKAE shows a steeper initial descent for LiH and H2O, indicating faster identification of effective gate sequences before plateauing.The passage attributes this behavior to the enhanced nonlinear expressivity of the QKAN layer.
  • B. Convergence Dynamics of Circuit Generation: C2H6 and the H2O dimer exhibit highly comparable convergence trajectories for GQE and GQKAE.For C2H6, GQE finds a marginally lower minimum, but both methods perform equivalently within statistical variance.
  • C. Results of Potential Energy Surfaces: Both generative models closely track CASCI dissociation profiles for H4, N2, and LiH, unlike HF, which deviates substantially under bond stretching.The comparisons include HF, CCSD, SCI, CASCI, and VQE reference methods.
  • C. Results of Potential Energy Surfaces: Discrete operator-pool optimization constructs highly entangled states that describe multireference character without continuous parameter tuning.This conclusion follows from the close agreement with CASCI across the dissociation coordinates.
  • C. Results of Potential Energy Surfaces: Using cc-pVDZ, both methods smoothly reproduce H2O angle-bending and water-dimer potential wells while describing orbital-symmetry shifts and weak intermolecular forces.The results attribute this flexibility to the generated discrete gate sequences.

D. Result of Absolute Error · E. Impact of Measurement Shots and Subspace Truncation · F. Quantum Resources and Parameter Efficiency

GQKAE and GQE maintain chemical-accuracy-level energy errors across the tested molecular regimes, while GQKAE reduces classical resource demands and improves circuit-generation speed. Measurement-shot and subspace-truncation studies show that accuracy improves with sufficient sampling and retained configurations.

  • D. Result of Absolute Error: Both GQKAE and GQE keep absolute errors within or below the 1.6 mHa chemical-accuracy threshold for H4, C2H6, and LiH across scanned geometries.Fig. 5 compares errors against CASCI on a logarithmic scale alongside HF, CCSD, and VQE reference baselines.
  • D. Result of Absolute Error: Between 1.0 ˚A and 2.0 ˚A in N2 dissociation, both models show significantly expanded standard deviations as the nitrogen triple bond breaks.The passage identifies this interval with strong static correlation and nearly degenerate electronic configurations.
  • D. Result of Absolute Error: For H2O and the water dimer with cc-pVDZ, both models remain below chemical accuracy across bending angles and close intermolecular distances.The reported robustness concerns high-order non-covalent interactions and dynamic correlations.
  • E. Impact of Measurement Shots and Subspace Truncation: 104 shots brings H2O energy error at 104.5◦ equilibrium angle below the chemical-accuracy threshold when dmax is fixed at 2000.The error steadily decreases as measurement shots increase.
  • E. Impact of Measurement Shots and Subspace Truncation: dmax = 100 achieves chemical accuracy for H2O when Nshots is fixed at 105, with performance plateauing beyond dmax = 1000.These results indicate that retaining critical electronic configurations is important for accuracy.
  • F. Quantum Resources and Parameter Efficiency: 66% is the approximate reduction in GQKAE trainable parameters relative to GQE, from roughly 42.7 M to 14.5 M.The reduction follows integration of the HQKAN module into the compressed latent space.
  • F. Quantum Resources and Parameter Efficiency: 6.8% to 16.7% are GQKAE circuit-generation wall-time speedups across tested molecules, while runtime memory falls from over 162 MB to ∼55 MB.GQKAE and GQE also have comparable gate requirements and achieve orders-of-magnitude reductions relative to the uncompiled UCCSD-VQE baseline.

VI. CONCLUSION

GQKAE achieves chemical accuracy while reducing classical resource demands and remaining robust under limited measurements and subspace truncation. Its compressed generative-model approach supports scalable HPC–quantum co-design, while current limitations include manually specified circuit length and evaluation on classical simulators.

  • Conclusion: 66% reduction in trainable parameters and memory relative to the GPT-2 baseline accompanies capture of complex chemical phenomena, including dynamic correlations and weak intermolecular forces.The result is reported for GQKAE relative to GPT-2.
  • Conclusion: GQKAE achieves chemical accuracy, computational speedups, and parameter reduction simultaneously under limited measurement shots and subspace truncation constraints.Circuit sequence length L currently must be specified manually.
  • Conclusion: Replacing parameterized quantum circuits with a classical generative model avoids barren-plateau trainability issues and generates significantly shallower circuits than standard VQE.The evaluations were performed using classical simulators, while reduced optimization difficulty and gate counts improve near-term device viability.
  • Conclusion: Compressed generative classical models can drive quantum-state preparation while reducing classical-side memory and runtime overhead for scalable HPC–quantum co-design.The QKAN-based backbone supports pretraining across families of Hamiltonians and fine-tuning for target systems.
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