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arxiv: 2605.15366 · v1 · pith:ZDISICJYnew · submitted 2026-05-14 · 🪐 quant-ph · physics.comp-ph

Measurement-Efficient Variational Quantum Linear Solver for Carleman-Linearized Nonlinear Dynamics

Yunya Liu , Pai Wang This is my paper

Pith reviewed 2026-05-19 15:23 UTC · model grok-4.3

classification 🪐 quant-ph physics.comp-ph
keywords variational quantum linear solverCarleman linearizationDuffing equationnonlinear dynamicshybrid quantum-classicalHadamard testquantum hardware implementation
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The pith

Variational quantum linear solvers recover states proportional to classical solutions for Carleman-linearized nonlinear dynamics.

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

The paper establishes a hybrid quantum-classical approach that first applies Carleman linearization to turn the weakly nonlinear Duffing equation into a solvable linear system, then uses an optimized variational quantum linear solver to find the corresponding quantum state. It reports that topology-agnostic ansatzes, careful Hermitianization choices, and efficient global or local cost functions let the solver reach near-unity fidelity and near-zero residuals on both simulated and real IBM and Xanadu hardware. A sympathetic reader would care because the method offers a concrete, measurement-efficient route to embed nonlinear dynamical simulation inside near-term quantum loops without demanding full error correction.

Core claim

Across block-banded test cases, VQLS with symmetry-grouped Hadamard Test evaluations under global and local cost formulations, together with comparisons of distinct Hermitianization methods and hardware-efficient ansatz architectures, achieves near-unity fidelity and vanishing relative residuals; these results demonstrate that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems.

What carries the argument

Topology-agnostic ansatz with optimized Hermitianization and efficient cost formulation inside the variational quantum linear solver applied to Carleman-linearized systems.

If this is right

  • Higher truncation orders in Carleman linearization produce successively better approximations to the original nonlinear dynamics.
  • Symmetry-grouped Hadamard tests support both global and local cost formulations while keeping measurement overhead low.
  • Multiple Hermitianization techniques can be evaluated inside one shared cost framework to identify the best performer for a given system.
  • Hardware-efficient ansatzes maintain high accuracy for block-banded Carleman matrices when Hermitianization is held fixed.

Where Pith is reading between the lines

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

  • The same pipeline could be tested on other nonlinear oscillators or fluid models that admit Carleman linearization.
  • Embedding the solver inside a feedback loop might enable real-time quantum-assisted prediction for weakly nonlinear control problems.
  • If the cost formulation scales favorably with matrix bandwidth, the approach could extend to larger truncation orders on future hardware.

Load-bearing premise

Carleman linearization accurately approximates the weakly nonlinear Duffing equation with errors that diminish as the truncation order increases.

What would settle it

An experiment in which raising the Carleman truncation order fails to reduce the gap between the VQLS output state and the classical solution vector, or in which fidelity remains low despite the listed optimizations, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.15366 by Pai Wang, Yunya Liu.

Figure 1
Figure 1. Figure 1: Hadamard Test circuits for the three overlap families required by VQLS: (a) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Verification of the Carleman linearization on the Duffing equation. (Left) Dis [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Global cost-function history for the Qiskit pipeline (Method A) on (a) a [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Heatmaps of the regularized system constructed by block-banded matrices [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Heatmaps of the augmented system constructed by real-valued block-banded [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 1
Figure 1. Figure 1: The symbolic form of matrix L where step m = 1, extended step p = 1, h is time step, δ is damping coefficient, α is linear stiffness coefficient, β is nonlinear stiffness coefficient, and γ is driving amplitude. 3 [PITH_FULL_IMAGE:figures/full_fig_p041_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Hadamard Test circuits for the imaginary part of expectation value mea [PITH_FULL_IMAGE:figures/full_fig_p042_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The commuting group strategy for a system with three target qubits and one [PITH_FULL_IMAGE:figures/full_fig_p043_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The block-banded matrix used in PennyLane Method A approach: (left) the [PITH_FULL_IMAGE:figures/full_fig_p044_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of different basis for two real-valued block-banded matrix in (a) and [PITH_FULL_IMAGE:figures/full_fig_p046_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cost history of examples of real-valued block-banded matrices generated by the [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Cost history and (b) heatmap of the highlighted example in Table III of the [PITH_FULL_IMAGE:figures/full_fig_p047_7.png] view at source ↗
read the original abstract

We present hybrid quantum-classical pipelines for solving the Duffing equation that leverage Carleman linearization and the Variational Quantum Linear Solver (VQLS). First, we demonstrate that Carleman linearization accurately approximates the weakly nonlinear Duffing equation, with errors diminishing as the truncation order increases. Next, across IBM and Xanadu platforms, we deploy VQLS with symmetry-grouped Hadamard Test evaluations under both global and local cost formulations, compare distinct Hermitianization within a common cost framework, and benchmark hardware-efficient ansatz architectures under a fixed Hermitianization. Across block-banded test cases, each method achieves near-unity fidelity and vanishing relative residuals. These results show that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems, providing a portable recipe for quantum-in-the-loop simulation of nonlinear dynamics.

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 presents hybrid quantum-classical pipelines for solving the Duffing equation via Carleman linearization to obtain a linear system, followed by solution with the Variational Quantum Linear Solver (VQLS). It first claims that Carleman linearization approximates the weakly nonlinear dynamics with errors that diminish as truncation order increases. It then reports hardware experiments on IBM and Xanadu platforms using symmetry-grouped Hadamard tests under global and local costs, comparisons of Hermitianization methods, and benchmarks of hardware-efficient ansatzes, all achieving near-unity fidelity and vanishing relative residuals on block-banded Carleman systems. The central claim is that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover states proportional to classical solutions for such systems.

Significance. If the results hold, the work is significant as a practical demonstration of VQLS applied to Carleman-linearized nonlinear ODEs on real hardware, providing a portable recipe that combines classical linearization with variational quantum linear solving. It contributes measurement-efficient techniques (symmetry-grouped Hadamard tests) and systematic comparisons of ansatz and Hermitianization choices within a common cost framework. The external grounding against classical solutions of the same linearized system strengthens the numerical claims.

major comments (2)
  1. [Abstract and numerical experiments] Abstract and results on Carleman approximation: the foundational claim that Carleman linearization approximates the weakly nonlinear Duffing equation with errors diminishing as truncation order increases is stated without explicit quantification or bounds for the specific truncation orders and block-banded matrix sizes used in the IBM/Xanadu VQLS experiments. Without showing that these linearization errors are smaller than the reported VQLS residuals, the validity of the hybrid pipeline for the original nonlinear dynamics is not secured.
  2. [Hardware experiments] Hardware results section: near-unity fidelity and vanishing relative residuals are reported without error bars, details on the number of circuit executions, or statistical analysis across runs. This omission makes it difficult to assess robustness given the noisy intermediate-scale quantum hardware used.
minor comments (1)
  1. [Methods] The description of the cost formulations and Hermitianization procedures would benefit from explicit equation references or a summary table to improve clarity for readers implementing the methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment point by point below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and numerical experiments] Abstract and results on Carleman approximation: the foundational claim that Carleman linearization approximates the weakly nonlinear Duffing equation with errors diminishing as truncation order increases is stated without explicit quantification or bounds for the specific truncation orders and block-banded matrix sizes used in the IBM/Xanadu VQLS experiments. Without showing that these linearization errors are smaller than the reported VQLS residuals, the validity of the hybrid pipeline for the original nonlinear dynamics is not secured.

    Authors: We agree that explicit quantification of the Carleman linearization error for the precise truncation orders and block-banded matrix dimensions employed in the hardware experiments, together with a direct comparison against the observed VQLS residuals, would better secure the validity of the overall pipeline. The manuscript already illustrates the general trend of diminishing error with increasing truncation order via classical numerical examples; however, these were not tied specifically to the VQLS test cases. In the revised version we will add a dedicated subsection (or supplementary figure) that computes and reports the linearization error norms for the exact truncation orders and matrix sizes used on IBM and Xanadu, and we will verify that these errors remain smaller than the reported VQLS residuals. revision: yes

  2. Referee: [Hardware experiments] Hardware results section: near-unity fidelity and vanishing relative residuals are reported without error bars, details on the number of circuit executions, or statistical analysis across runs. This omission makes it difficult to assess robustness given the noisy intermediate-scale quantum hardware used.

    Authors: We acknowledge that the absence of error bars, shot counts, and statistical analysis across multiple runs limits the reader’s ability to judge robustness on NISQ devices. In the revised manuscript we will augment the hardware-results section with (i) error bars obtained from repeated independent executions, (ii) explicit reporting of the number of circuit shots used for each Hadamard-test evaluation, and (iii) a concise statistical summary (mean and standard deviation) of fidelity and residual metrics over the ensemble of runs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims grounded in external classical benchmarks

full rationale

The paper separates two independent demonstrations: (1) classical verification that Carleman truncation approximates the weakly nonlinear Duffing equation with errors that decrease at higher orders, and (2) VQLS applied to the resulting block-banded linear systems, with quantum outputs directly compared to classical solutions of the exact same Carleman-linearized matrices. Fidelity and residual metrics are therefore measured against an external classical reference rather than against any fitted parameter or self-derived quantity internal to the VQLS procedure. No equation or claim reduces by construction to its own inputs; the topology-agnostic ansatz, Hermitianization choices, and cost formulations are validated by their agreement with independently computed classical vectors. This structure satisfies the self-contained benchmark criterion and yields no load-bearing self-definition, fitted-input prediction, or self-citation chain.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that Carleman truncation converges for weakly nonlinear systems and on standard variational quantum algorithm components; no new physical entities are introduced.

free parameters (2)
  • Carleman truncation order
    Selected to balance approximation error against matrix size for the Duffing equation; value not specified in abstract.
  • VQLS ansatz depth and parameters
    Optimized during variational training; specific circuit details absent from abstract.
axioms (1)
  • domain assumption Carleman linearization provides a controllable approximation to the nonlinear Duffing dynamics whose error decreases with truncation order
    Invoked in the first sentence of the abstract as the basis for the subsequent quantum solver experiments.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We present hybrid quantum-classical pipelines for solving the Duffing equation that leverage Carleman linearization and the Variational Quantum Linear Solver (VQLS). ... topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Carleman linearization accurately approximates the weakly nonlinear Duffing equation, with errors diminishing as the truncation order increases

What do these tags mean?
matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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