arxiv: 2604.10099 · v1 · submitted 2026-04-11 · 🪐 quant-ph

Recognition: unknown

Quantum Error Mitigation Strategies for Variational PDE-Constrained Circuits on Noisy Hardware

Prasad Nimantha Madusanka Ukwatta Hewage , Midhun Chakkravarthy , Ruvan Kumara Abeysekara

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:41 UTC · model grok-4.3

classification 🪐 quant-ph

keywords variational quantum circuitsquantum error mitigationpartial differential equationsnoise resiliencephysics-informed constraintszero-noise extrapolationNISQ hardware

0 comments

The pith

Physics-constrained variational circuits maintain 25-47% higher fidelity under noise than unconstrained ones when solving PDEs.

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

The paper tests variational quantum circuits for solving partial differential equations on noisy hardware and compares three error mitigation approaches. It shows that constraining the circuits with PDE residual losses builds in noise resilience that grows with the complexity of the equation. At a noise strength of 0.01, these constrained circuits hold 25-47% higher fidelity than unconstrained versions across depolarizing, amplitude damping, and bit-flip channels. The work matters because it offers a route to usable accuracy on current NISQ devices without requiring perfect hardware or heavy post-processing. Tests also reveal that zero-noise extrapolation cuts absolute errors by 82-96% at low noise but that systematic errors still dominate the budget at all noise levels.

Core claim

Variational quantum circuits constrained by PDE residual loss functions exhibit inherent noise resilience. This is established analytically and confirmed numerically for the heat equation, Burgers' equation, and Saint-Venant shallow water equations under three noise channels. At p = 0.01, constrained circuits achieve 25-47% higher fidelity than unconstrained counterparts, with the advantage scaling with PDE complexity. Zero-noise extrapolation reduces absolute error by 82-96% at p = 0.001, while error budget analysis shows systematic errors at 43-58% and the PDE residual error share rising from 10% to 31% as noise increases.

What carries the argument

PDE residual loss functions that constrain the variational circuits and supply structured gradient information to absorb noise effects.

Load-bearing premise

The chosen noise channels and 6-qubit 4-layer circuit sizes represent the noise and depths that will appear in future practical variational PDE solvers on NISQ hardware.

What would settle it

Running the same constrained versus unconstrained fidelity comparison on real NISQ hardware at p around 0.01, or scaling the experiments beyond 6 qubits and 4 layers, would directly test whether the reported resilience holds outside simulation.

Figures

Figures reproduced from arXiv: 2604.10099 by Midhun Chakkravarthy, Prasad Nimantha Madusanka Ukwatta Hewage, Ruvan Kumara Abeysekara.

**Figure 2.** Figure 2: FIG. 2. ZNE effectiveness across noise models. Solid lines: mitigated error; dashed lines with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Left: PEC sampling overhead vs. gate count for different noise levels. Right: Accuracy [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Physics-constrained (green) vs. unconstrained (red) fidelity under depolarizing noise. The [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Training convergence under noise. Dashed: noisy without mitigation. Solid colored: with [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Error budget decomposition. Systematic errors (red) dominate; the PDE residual compo [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

read the original abstract

Variational quantum circuits (VQCs) solving partial differential equations (PDEs) on near-term quantum hardware face a critical challenge: hardware noise degrades solution fidelity and disrupts convergence. We present a systematic study of three noise channels; depolarizing, amplitude damping, and bit-flip on VQCs constrained by PDE residual loss functions for the heat equation, Burgers' equation, and the Saint-Venant shallow water equations. We benchmark three error mitigation strategies: zero-noise extrapolation (ZNE) via Richardson polynomial fitting, probabilistic error cancellation (PEC), and measurement error mitigation through inverse confusion matrices. Our numerical experiments on 6-qubit, 4-layer circuits demonstrate that ZNE reduces absolute error by 82-96% at low noise (p = 0.001), with effectiveness degrading gracefully at higher noise strengths. We prove analytically and confirm numerically that physics-constrained circuits exhibit inherent noise resilience: at p = 0.01, constrained circuits maintain 25-47% higher fidelity than unconstrained counterparts, with the advantage scaling with PDE complexity. PEC provides near-exact correction at low gate counts but incurs exponential sampling overhead, rendering it impractical beyond ~60 gates at p >= 0.02. Error budget decomposition reveals that systematic errors dominate at all noise levels (43-58%), while the PDE residual component grows from ~10% to ~31% as noise increases, indicating that physics constraints absorb noise through structured gradient information. These results establish practical guidelines for deploying variational PDE solvers on NISQ hardware.

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.

Desk Editor's Note private letter to a colleague

PDE constraints add measurable noise resilience in small VQCs, with concrete error budgets from the benchmarks but limited to toy scales.

read the letter

The main thing to know is that this paper shows physics-constrained variational circuits for PDEs keep 25-47% higher fidelity than unconstrained ones at 1% noise, and they map out how ZNE, PEC, and measurement mitigation trade off on 6-qubit circuits for the heat, Burgers, and Saint-Venant equations. The error budget split is the most useful part: systematic errors dominate at 43-58% across noise levels, while the residual term grows to 31% as noise increases, suggesting the constraints channel some noise into structured gradients rather than random drift. ZNE cuts absolute error 82-96% at low noise but degrades at higher strengths, and PEC hits near-exact correction only below about 60 gates before sampling costs explode. That decomposition gives practical rules of thumb for NISQ deployment. The analytical resilience argument is new in this combined setting, and the numerical confirmation on multiple PDEs is straightforward and internally consistent for the stated depolarizing, damping, and bit-flip channels. The work is narrow but honest about its scope. The soft spots are the usual ones for this scale: the proof steps are asserted rather than walked through, all results sit on 6-qubit 4-layer circuits with no error bars or statistical tests reported, and the noise models are textbook rather than device-specific. Whether the fidelity advantage holds at deeper circuits or on real hardware remains open. This is for people already running variational PDE solvers on current hardware who need concrete mitigation guidelines. A reader in quantum scientific computing or error mitigation would pull usable numbers and trade-offs from it. It deserves a serious referee because the benchmarking is systematic and the error analysis is reproducible in principle, even if the claims need tighter documentation on the analytics and implementation details. Send it for review after they expand the proof and add reproducibility notes.

Referee Report

3 major / 3 minor

Summary. The manuscript studies error mitigation for variational quantum circuits (VQCs) solving PDEs on noisy hardware. It examines depolarizing, amplitude damping, and bit-flip noise on PDE-residual-constrained VQCs for the heat, Burgers', and Saint-Venant equations. Three mitigation strategies (ZNE via Richardson extrapolation, PEC, and measurement error mitigation) are benchmarked on 6-qubit 4-layer circuits. The central claims are an analytical proof of inherent noise resilience in physics-constrained circuits (with 25-47% higher fidelity at p=0.01, scaling with PDE complexity) plus numerical confirmation showing ZNE reduces absolute error by 82-96% at low noise, PEC is effective only at low gate counts, and systematic errors dominate the error budget (43-58%).

Significance. If the results hold, the work provides practical guidelines for NISQ deployment of variational PDE solvers by showing that physics constraints confer noise resilience and by quantifying mitigation trade-offs. The error-budget decomposition (systematic vs. PDE-residual components) and the scaling of advantage with PDE complexity are useful contributions. Credit is due for combining an analytical argument with multi-PDE numerical benchmarking; however, the absence of machine-checked proofs or fully reproducible code limits the strength of the reproducibility claim.

major comments (3)

[Numerical Experiments] Numerical Experiments section: the 6-qubit 4-layer circuit implementations (ansatz, gate set, optimizer, and exact procedure for evaluating the noisy PDE residual loss) are unspecified. This directly affects reproducibility of the reported 82-96% error reduction and the 25-47% fidelity advantage, which are load-bearing for the numerical confirmation of the central resilience claim.
[Abstract] Abstract and Results section: the manuscript asserts an 'analytical proof' of inherent noise resilience without outlining the key derivation steps or citing the specific section/equation (e.g., the loss-function analysis or gradient-structure argument) where it appears. Because this proof is presented as foundational to the 25-47% fidelity advantage, its explicit location and main steps must be provided for verification.
[Results] Results on fidelity comparisons: the exact percentages (25-47% higher fidelity at p=0.01, scaling with PDE complexity) are stated without error bars, number of independent runs, or statistical tests. Given the stochastic nature of noisy circuit simulations, this weakens support for the quantitative claims that underpin the noise-resilience conclusion.

minor comments (3)

[Abstract] The abstract lists three mitigation strategies but gives only a one-sentence description of measurement error mitigation; a brief expansion would improve clarity.
[Results] A summary table comparing fidelity, error reduction, and overhead across the three PDEs and noise strengths would help readers quickly assess the scaling claims.
[Introduction] The noise-strength parameter p is introduced without an early explicit definition of the channel (e.g., depolarizing probability per gate); consistent early notation would aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's thorough evaluation of our manuscript on quantum error mitigation for variational PDE-constrained circuits. The comments highlight important areas for improving clarity and reproducibility, which we address below. We have prepared revisions to incorporate the suggested changes where appropriate.

read point-by-point responses

Referee: Numerical Experiments section: the 6-qubit 4-layer circuit implementations (ansatz, gate set, optimizer, and exact procedure for evaluating the noisy PDE residual loss) are unspecified. This directly affects reproducibility of the reported 82-96% error reduction and the 25-47% fidelity advantage, which are load-bearing for the numerical confirmation of the central resilience claim.

Authors: We agree that the lack of specific implementation details in the Numerical Experiments section hinders reproducibility. In the revised manuscript, we will provide a comprehensive description of the circuit ansatz, including the specific parameterized gates and their arrangement in the 4-layer structure; the gate set employed (e.g., single-qubit rotations and entangling gates); the optimizer used (including hyperparameters such as learning rate and convergence criteria); and the precise procedure for computing the noisy PDE residual loss, detailing how noise channels are simulated and applied to the circuit evaluations. These additions will allow readers to replicate the numerical results exactly. revision: yes
Referee: Abstract and Results section: the manuscript asserts an 'analytical proof' of inherent noise resilience without outlining the key derivation steps or citing the specific section/equation (e.g., the loss-function analysis or gradient-structure argument) where it appears. Because this proof is presented as foundational to the 25-47% fidelity advantage, its explicit location and main steps must be provided for verification.

Authors: The analytical proof is detailed in Section 3 of the manuscript, focusing on the loss-function analysis and the gradient structure under noise. However, we acknowledge that the abstract and results do not sufficiently highlight or outline the key steps. In the revision, we will update the abstract to reference Section 3 explicitly and include a concise outline of the main derivation steps: starting from the PDE residual loss function, demonstrating how the constraint leads to noise-resilient gradients, and deriving the fidelity advantage scaling with PDE complexity. This will facilitate verification of the proof. revision: yes
Referee: Results on fidelity comparisons: the exact percentages (25-47% higher fidelity at p=0.01, scaling with PDE complexity) are stated without error bars, number of independent runs, or statistical tests. Given the stochastic nature of noisy circuit simulations, this weakens support for the quantitative claims that underpin the noise-resilience conclusion.

Authors: We recognize the importance of statistical rigor in reporting the fidelity comparisons, given the stochastic elements in noisy simulations. The percentages were derived from ensemble averages, but error bars, the number of independent runs (typically 50-100 per configuration), and any relevant statistical tests were omitted. We will revise the Results section to include these details, such as mean fidelity values with standard deviations and confirmation of statistical significance where applicable, to better support the quantitative claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its central claims of inherent noise resilience and mitigation performance through direct analytical proofs of constrained loss structures plus numerical benchmarking on 6-qubit circuits against standard external noise models (depolarizing, amplitude damping, bit-flip). These steps do not reduce by construction to fitted parameters, self-definitions, or self-citation chains; the reported fidelity advantages, error budgets, and mitigation effectiveness follow from explicit comparisons and decompositions that remain falsifiable against the stated PDEs and circuit depths. No load-bearing uniqueness theorems or ansatzes imported from prior author work are invoked in the abstract or context.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard quantum noise models and the assumption that PDE residual loss functions can be evaluated on noisy hardware; no new entities or ad-hoc parameters are introduced beyond the experimental noise strengths.

axioms (1)

domain assumption Standard depolarizing, amplitude-damping, and bit-flip channels accurately capture dominant hardware noise for the studied gate sets.
Invoked when applying the three noise channels to the VQCs in the numerical experiments.

pith-pipeline@v0.9.0 · 5594 in / 1258 out tokens · 37439 ms · 2026-05-10T16:41:01.272832+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Variational quantum algorithms,

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, “Variational quantum algorithms,”Nat. Rev. Phys.3, 625–644 (2021)

2021
[2]

Noisy intermediate-scale quantum algorithms,

K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru- Guzik, “Noisy intermediate-scale quantum algorithms,”Rev. Mod. Phys.94, 015004 (2022)

2022
[3]

Variational quantum algorithms for nonlinear problems,

M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, “Variational quantum algorithms for nonlinear problems,”Phys. Rev. A101, 010301(R) (2020)

2020
[4]

Solving nonlinear differential equations with differentiable quantum circuits,

O. Kyriienko, A. E. Paine, and V. E. Elfving, “Solving nonlinear differential equations with differentiable quantum circuits,”Phys. Rev. A103, 052416 (2021)

2021
[5]

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial dif- ferential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial dif- ferential equations,”J. Comput. Phys.378, 686–707 (2019)

2019
[6]

Schuld and F

M. Schuld and F. Petruccione,Machine Learning with Quantum Computers(Springer, Cham, 2021), 2nd ed

2021
[7]

Quantum Computing in the NISQ era and beyond,

J. Preskill, “Quantum Computing in the NISQ era and beyond,”Quantum2, 79 (2018). 15

2018
[8]

Noise- induced barren plateaus in variational quantum algorithms,

S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, “Noise- induced barren plateaus in variational quantum algorithms,”Nat. Commun.12, 6961 (2021)

2021
[9]

The adjoint is all you need: Characterizing barren plateaus in quantum ans¨ atze,

E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cirstoiu, “The adjoint is all you need: Characterizing barren plateaus in quantum ans¨ atze,” arXiv:2309.07902 (2023)

work page arXiv 2023
[10]

Error mitigation for short-depth quantum cir- cuits,

K. Temme, S. Bravyi, and J. M. Gambetta, “Error mitigation for short-depth quantum cir- cuits,”Phys. Rev. Lett.119, 180509 (2017)

2017
[11]

Practical quantum error mitigation for near-future applications,

S. Endo, S. C. Benjamin, and Y. Li, “Practical quantum error mitigation for near-future applications,”Phys. Rev. X8, 031027 (2018)

2018
[12]

Quantum error mitigation,

Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, “Quantum error mitigation,”Rev. Mod. Phys.95, 045005 (2023)

2023
[13]

Efficient variational quantum simulator incorporating active error minimization,

Y. Li and S. C. Benjamin, “Efficient variational quantum simulator incorporating active error minimization,”Phys. Rev. X7, 021050 (2017)

2017
[14]

Mitigating measure- ment errors in multiqubit experiments,

S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, “Mitigating measure- ment errors in multiqubit experiments,”Phys. Rev. A103, 042605 (2021)

2021
[15]

Error mitigation extends the computational reach of a noisy quantum processor,

A. Kandala, K. Temme, A. D. C´ orcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, “Error mitigation extends the computational reach of a noisy quantum processor,”Nature 567, 491–495 (2019)

2019
[16]

Evidence for the utility of quantum computing before fault tolerance,

Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, “Evidence for the utility of quantum computing before fault tolerance,”Nature618, 500–505 (2023)

2023
[17]

The theory of variational hybrid quantum-classical algorithms,

J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,”New J. Phys.18, 023023 (2016)

2016
[18]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cam- bridge University Press, Cambridge, 2010), 10th anniversary ed

2010
[19]

PennyLane: Automatic differentiation of hybrid quantum-classical computations

V. Bergholmet al., “PennyLane: Automatic differentiation of hybrid quantum-classical com- putations,” arXiv:1811.04968 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[20]

Quantum circuit learning,

K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, “Quantum circuit learning,”Phys. Rev. A98, 032309 (2018)

2018
[21]

Evaluating analytic gradients on quantum hardware,

M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, “Evaluating analytic gradients on quantum hardware,”Phys. Rev. A99, 032331 (2019)

2019
[22]

Barren plateaus in 16 quantum neural network training landscapes,

J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, “Barren plateaus in 16 quantum neural network training landscapes,”Nat. Commun.9, 4812 (2018)

2018
[23]

Cost function dependent barren plateaus in shallow parametrized quantum circuits,

M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits,”Nat. Commun.12, 1791 (2021)

2021
[24]

A variational eigenvalue solver on a photonic quantum processor,

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor,”Nat. Commun.5, 4213 (2014)

2014
[25]

A Quantum Approximate Optimization Algorithm

E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” arXiv:1411.4028 (2014)

work page internal anchor Pith review Pith/arXiv arXiv 2014
[26]

Parameterized quantum circuits as machine learning models,

M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, “Parameterized quantum circuits as machine learning models,”Quantum Sci. Technol.4, 043001 (2019)

2019
[27]

Fundamental limits of quantum error mitigation,

R. Takagi, S. Endo, S. Minagawa, and M. Murao, “Fundamental limits of quantum error mitigation,”npj Quantum Inf.8, 114 (2022)

2022
[28]

Universal cost bound of quantum error mitigation based on quantum estimation theory,

K. Tsubouchi, T. Sagawa, and N. Yoshioka, “Universal cost bound of quantum error mitigation based on quantum estimation theory,”Phys. Rev. Lett.131, 210601 (2023)

2023
[29]

Quantum error mitigation as a universal error reduction technique: Applications from the NISQ to the fault-tolerant quantum computing eras,

Y. Suzuki, S. Endo, K. Fujii, and Y. Tokunaga, “Quantum error mitigation as a universal error reduction technique: Applications from the NISQ to the fault-tolerant quantum computing eras,”PRX Quantum3, 010345 (2022)

2022
[30]

Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors,

E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, “Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors,”Nat. Phys.19, 1116–1121 (2023)

2023
[31]

Exponential error suppression for near-term quantum devices,

B. Koczor, “Exponential error suppression for near-term quantum devices,”Phys. Rev. X11, 031057 (2021)

2021
[32]

Virtual distillation for quantum error mitigation,

W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, “Virtual distillation for quantum error mitigation,”Phys. Rev. X11, 041036 (2021)

2021
[33]

Error mitigation with Clifford quantum- circuit data,

P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, “Error mitigation with Clifford quantum- circuit data,”Quantum5, 592 (2021)

2021
[34]

Unified approach to data-driven quantum error mitigation,

A. Lowe, M. H. Gordon, P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, “Unified approach to data-driven quantum error mitigation,”Phys. Rev. Research3, 033098 (2021)

2021
[35]

Learning-based quantum error mitigation,

A. Strikis, D. Qin, Y. Chen, S. C. Benjamin, and Y. Li, “Learning-based quantum error mitigation,”PRX Quantum2, 040330 (2021). 17

2021
[36]

Sampling overhead analysis of quantum error mitigation: Uncoded vs. coded systems,

Y. Xiong, D. Chandra, S. X. Ng, and L. Hanzo, “Sampling overhead analysis of quantum error mitigation: Uncoded vs. coded systems,”IEEE Access8, 228967–228991 (2020). 18

2020