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arxiv: 2412.03939 · v3 · submitted 2024-12-05 · 🪐 quant-ph · cs.CE

A quantum nonlinear solver based on the asymptotic numerical method

Yongchun Xu , Zengtao Kuang , Qun Huang , Jie Yang , Hamid Zahrouni , Michel Potier-Ferry , Kaixuan Huang , Jia-Chi Zhang
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Heng Fan Heng Hu
This is my paper

Pith reviewed 2026-05-23 08:05 UTC · model grok-4.3

classification 🪐 quant-ph cs.CE
keywords quantum asymptotic numerical methodnonlinear solvervariational quantum linear solversuperconducting quantum processorTaylor series expansionquantum-enhanced Jacobi methodnonlinear computational mechanicsperturbation techniques
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The pith

The quantum asymptotic numerical method converts nonlinear problems into sequences of linear equations solved on quantum hardware, demonstrated by 98% accuracy in a superconducting processor experiment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents the quantum asymptotic numerical method (qANM) for solving nonlinear problems by applying high-order Taylor series expansions to generate sequences of linear equations. These linear systems are then addressed using the variational quantum linear solver integrated with a quantum-enhanced Jacobi method. The approach is validated through numerical simulations on a quantum simulator showing convergence, and crucially through a proof-of-principle experiment on a superconducting quantum processor that achieves 98% accuracy in tracking the nonlinear solution path despite inherent noise. A sympathetic reader cares because this offers a pathway for quantum computing to tackle nonlinear systems prevalent in computational mechanics and engineering.

Core claim

By basing the solver on high-order perturbation techniques and Taylor series expansions, the qANM transforms complex nonlinear systems into manageable sequences of linear equations that can be solved quantumly, with the variational quantum linear solver and quantum-enhanced Jacobi method enabling the process, as confirmed by simulator validations and an experimental run reaching 98% accuracy on noisy hardware.

What carries the argument

The qANM framework that uses high-order Taylor series expansions to linearize nonlinear problems, solved via variational quantum linear solver combined with quantum-enhanced Jacobi method.

If this is right

  • The high-order ANM formulation captures the solution path effectively through Taylor series expansions.
  • Integration with variational quantum linear solver allows handling of the resulting linear systems on quantum devices.
  • Quantum-enhanced Jacobi method contributes to solving the sequence of equations.
  • The method demonstrates robustness for nonlinear problems on near-term quantum hardware.
  • Numerical simulations confirm the convergence of the method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may scale to more complex nonlinear problems in mechanics if noise can be further managed.
  • Similar perturbation-based methods could be adapted for other quantum algorithms beyond linear solvers.
  • Success on superconducting processors suggests potential applicability to other quantum hardware platforms.

Load-bearing premise

The high-order Taylor series expansions combined with the variational quantum linear solver and quantum-enhanced Jacobi method will remain stable and converge on noisy quantum hardware without requiring error mitigation techniques beyond those implicitly assumed.

What would settle it

An experiment repeating the proof-of-principle on the superconducting quantum processor that fails to achieve high accuracy in tracking the nonlinear solution path or shows divergence in the Taylor series expansions.

Figures

Figures reproduced from arXiv: 2412.03939 by Hamid Zahrouni, Heng Fan, Heng Hu, Jia-Chi Zhang, Jie Yang, Kaixuan Huang, Michel Potier-Ferry, Qun Huang, Yongchun Xu, Zengtao Kuang.

Figure 1
Figure 1. Figure 1: (a) Illustration of the hardware-efficient ansatz [ [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for computing the inner product between a row of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance comparison of VQLS and q-Jacobi in solving [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Accuracy of solutions obtained by VQLS and q-Jacobi for different right-hand side vectors [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Impact of the number of shots ns on the performance of q-Jacobi for solving Ku = F0. (a) Accuracy of the solution versus ns, with the classical Jacobi method accuracy as a reference (blue dashed line). (b) Number of iterations required for convergence versus ns, compared to the classical Jacobi method (blue dashed line). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of VQLS and q-Jacobi in terms of the number of executed quantum circuits [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic of the spring-mass problem [71] [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Solution paths (w1, λ) obtained by qANM combined with VQLS, qANM combined with q-Jacobi, and the NR method combined with q-Jacobi, along with the reference analytical solution. (b) Relative error of w1 for each method compared to the analytical solution. 0 1 2 3 4 Load 6 (N) (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 P o sitio n w1 (m m) Reference qANM, shots ns = 10 1 qANM, shots ns = 10 2 qANM, shots ns = 10 5 … view at source ↗
Figure 9
Figure 9. Figure 9: (a) Solution paths obtained by qANM with different numbers of shots [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Sketch of the Euler-Bernoulli beam, where the upper surface is applied with uniform pressure. [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Displacement w at the midpoint of the Euler-Bernoulli beam, comparing results obtained using the proposed qANM and the reference classical NR method. E = 3 × 105 MPa. The governing equations for the problem are as follows: ϵxx = ∂u ∂x + 1 2  ∂w ∂x 2 − z ∂ 2w ∂x2 , (23a) σxx = Eϵxx, (23b) Z V δϵxxσxxdV − Z L 0 δwλq0Bdx = 0 (23c) where ϵxx is the von K´arm´an strain, σxx is the stress, w represents the di… view at source ↗
Figure 12
Figure 12. Figure 12: Stress distribution σxx along the Euler-Bernoulli beam at the final solution point, comparing results from the qANM and the reference computed by the classical NR method. Note that the height of the displayed beam has been magnified by a factor of 5 for better visualization. are used for this problem [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Quantum circuit for solving the spring-mass problem using qANM, executed on the Quafu [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Results of solving a linear equation using q-Jacobi on the Quafu quantum processor: evolution [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Results for the spring-mass problem on the Quafu quantum processor: (a) Solution paths for [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Probability P0 obtained from the Quafu quantum processor. (a) True P0 versus the estimated P0 on the Quafu quantum processor, along with the correlation coefficient R. (b) Comparison of the absolute error in the estimated P0 obtained on the Quafu quantum processor and the Qiskit simulator, both with the same number of shots ns = 5 × 104 . Finally, to evaluate the influence of quantum hardware noise, [PIT… view at source ↗
Figure 17
Figure 17. Figure 17: Sketch of the Euler-Bernoulli beam, where the boundary condition triggers buckling with [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Displacement w at the midpoint of the Euler-Bernoulli beam, obtained using ANM with the multiple-linear-solving scheme and the NR method [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (a) Convergence of the cost function C(θ) during optimization iterations for VQMI. (b) Comparison of the computed matrix inverse K−1 with the reference. proposed VQMI. In such a scenario, integrating VQMI into qANM could further exploit the benefits of both ANM and quantum computing. 7 Conclusion In this work, we have developed a new quantum nonlinear solver called qANM, which offers a large convergence r… view at source ↗
read the original abstract

Quantum computing offers a promising avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method (qANM), a framework for solving nonlinear problems using quantum computing. Based on the principle of high-order perturbation techniques, the proposed method uses Taylor series expansions to transform complex nonlinear systems into sequences of linear equations. We integrate the method with the variational quantum linear solver and a quantum-enhanced Jacobi method. Numerical simulations on a quantum simulator validate the convergence of the method. In particular, the high-order ANM formulation demonstrates robustness in addressing nonlinear problems by effectively capturing the solution path through Taylor series expansions. Furthermore, a highlight of this work is a proof-of-principle experiment on a superconducting quantum processor. Despite the noise inherent in near-term quantum hardware, the experiment achieves 98% accuracy in tracking the nonlinear solution path. We believe this work provides a useful reference for applying quantum computing to nonlinear computational mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the quantum asymptotic numerical method (qANM), which applies high-order Taylor series expansions to convert nonlinear problems into sequences of linear systems solved using the variational quantum linear solver (VQLS) and a quantum-enhanced Jacobi method. Validation is reported via numerical simulations on a quantum simulator demonstrating convergence, together with a proof-of-principle experiment on a superconducting quantum processor that achieves 98% accuracy in tracking the nonlinear solution path despite hardware noise.

Significance. If the experimental result holds, the work would constitute an early experimental bridge from quantum linear solvers to nonlinear problems in computational mechanics. The perturbation-based reduction to linear systems is a standard technique that aligns well with existing VQLS capabilities and could scale to larger nonlinear analyses if noise robustness is confirmed.

major comments (2)
  1. [Experimental Results] Experimental section: The central claim of 98% accuracy on a superconducting processor is load-bearing for the manuscript's contribution, yet no information is supplied on qubit count, circuit depth, ansatz choice, noise model, or error mitigation techniques. Without these details it is impossible to determine whether the reported accuracy reflects genuine robustness of the qANM+VQLS pipeline or depends on unstated low-noise conditions or post-processing.
  2. [Method Description] Method and convergence analysis: The high-order Taylor expansion is asserted to capture the solution path robustly, but no quantitative bound or numerical study is given on truncation error accumulation when each linear system is solved approximately by VQLS on noisy hardware. This directly affects the validity of the convergence claims made for both the simulator and the hardware experiment.
minor comments (1)
  1. [Abstract] The abstract states that simulations 'validate the convergence of the method' but does not identify the specific nonlinear test problem or the order of the Taylor expansion used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback, which helps strengthen the manuscript. We address each major comment below and will revise the paper to incorporate the requested details and analysis.

read point-by-point responses

  1. Referee: [Experimental Results] Experimental section: The central claim of 98% accuracy on a superconducting processor is load-bearing for the manuscript's contribution, yet no information is supplied on qubit count, circuit depth, ansatz choice, noise model, or error mitigation techniques. Without these details it is impossible to determine whether the reported accuracy reflects genuine robustness of the qANM+VQLS pipeline or depends on unstated low-noise conditions or post-processing.

    Authors: We agree that these implementation details are essential for evaluating the experimental result. In the revised manuscript we will add the qubit count, circuit depths, ansatz choice for VQLS, the noise model of the superconducting device, and any error-mitigation steps employed. The 98 % figure was obtained on the specific hardware configuration that will now be fully documented, allowing readers to assess the conditions under which the accuracy was achieved. revision: yes

  2. Referee: [Method Description] Method and convergence analysis: The high-order Taylor expansion is asserted to capture the solution path robustly, but no quantitative bound or numerical study is given on truncation error accumulation when each linear system is solved approximately by VQLS on noisy hardware. This directly affects the validity of the convergence claims made for both the simulator and the hardware experiment.

    Authors: We acknowledge the absence of a quantitative error-propagation study. While simulator results show convergence, we did not analyze how VQLS approximation errors accumulate across the Taylor orders on noisy hardware. The revised manuscript will include a dedicated numerical study of truncation-error accumulation under approximate linear solves, together with a discussion of its implications for the reported convergence on both simulator and hardware. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard perturbation theory and external quantum solvers

full rationale

The paper presents qANM as a combination of high-order Taylor series expansions (standard asymptotic numerical method) with VQLS and a quantum-enhanced Jacobi method. The 98% accuracy result is an experimental outcome on superconducting hardware, not a fitted parameter or self-referential prediction. No equations or steps in the provided abstract reduce the central claim to its own inputs by construction, and no load-bearing self-citations or uniqueness theorems are invoked. The method chain is self-contained against external benchmarks from classical ANM and variational quantum algorithms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review limited to abstract; the approach rests on standard mathematical assumptions of Taylor series convergence for nonlinear functions and the applicability of variational quantum linear solvers to the resulting sequence of linear systems.

axioms (2)
  • standard math Taylor series expansions can be used to transform nonlinear systems into sequences of linear equations
    Invoked in the abstract to convert complex nonlinear problems into solvable linear steps.
  • domain assumption Variational quantum linear solver and quantum-enhanced Jacobi method can be integrated with the perturbation sequence
    Stated as the integration basis for solving the generated linear equations on quantum hardware.

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