Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation

Chenghua Duan; Jungang Xu; Wanqi Sun

REVIEW 5 major objections 4 minor 24 references

Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation

T0 review · 5 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A matrix-product-state circuit plus neural zero-noise extrapolation keeps the noisy H4 ground-state energy within 0.017 Hartree of exact, while conventional VQE variants deviate by up to 1.6 Hartree.

desk verdict A plausible but under-validated MPS-VQE+ZNE combination; the central noisy energy figure may be an extrapolation artifact. read the letter →

arxiv 2501.01646 v2 pith:TDD5ZKLG submitted 2025-01-03 quant-ph

Wanqi Sun , Jungang Xu , Chenghua Duan This is my paper · ORCID

classification quant-ph

keywords variationalquantumeigensolvermatrixproductstateszero-noiseextrapolationneuralnetworkerrormitigationchemistryH4moleculenoisyintermediate-scalePaulimeasurementgrouping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that variational quantum eigensolvers can deliver accurate molecular ground-state energies on near-term noisy hardware rather than only in ideal noiseless simulation. It constructs the trial state as a matrix product state, maps that structure into a shallow parameterized circuit, pre-trains the circuit parameters classically, and applies zero-noise extrapolation in which a small neural network fits expectation values at several elevated noise levels and evaluates the fit at zero noise. On an eight-qubit simulation of H4 with depolarizing, thermal relaxation, and bit-flip noise, the method reports a noisy ground-state energy of -2.1490 Hartree against the full-configuration-interaction value of -2.1664 Hartree, an error of about 0.017 Hartree, while four mainstream VQE variants land between -0.5293 and -1.6726 Hartree. The significance of the claim is that a compact circuit plus error mitigation, rather than a very deep circuit, may be enough for useful quantum chemistry on today's devices.

What carries the argument

The central object is the matrix-product-state ansatz treated as a parameterized quantum circuit: each local MPS tensor is mapped to a two-qubit unitary of the form $(U(\theta_{n,l_0})\otimes U(\theta_{n,l_1}))\,\mathrm{CNOT}\,(U(\theta_{n,r_0})\otimes U(\theta_{n,r_1}))$, with single-qubit $U(\theta_*)=R_Z(\theta_0)R_Y(\theta_1)R_Z(\theta_2)$, laid out in a brick-wall pattern so the full eight-qubit circuit has 91 gates and 84 parameter gates. The argument is carried by three mechanisms: center-orthogonal gauging of the MPS, which stabilizes local tensor updates and simplifies energy contraction; classical pre-training of the tensor parameters so the circuit starts near the solution and avoids initialization-driven fluctuations; and zero-noise extrapolation, where circuit folding scales the noise and a three-layer fully connected neural network fits the expectation values at nonzero noise levels to extrapolate the value at zero noise. Commuting Pauli strings of the Hamiltonian are grouped so that compatible terms are measured in the same shot, reducing sampling overhead.

What would settle it

Run the same H4 simulation under the same depolarizing, thermal-relaxation, and bit-flip noise model, but replace the neural-network fit with a standard polynomial or exponential zero-noise extrapolation on the same folded circuits; if the polynomial-extrapolated energy differs from the reported -2.1490 Hartree by more than the claimed $\mathcal{O}(10^{-2})$ error, then the neural extrapolator, rather than the circuit or pre-training, is the source of the improvement.

Watch

Extended reading notes

Core claim

The central claim is that combining an MPS-structured ansatz with classical pre-training and neural-network zero-noise extrapolation makes VQE markedly more noise-robust. In noiseless simulation the 91-gate circuit reaches -2.1609 Hartree, close to the UCCSD value of -2.1615 Hartree and to the FCI benchmark of -2.1664 Hartree, despite using roughly thirty times fewer gates than UCCSD. Under the specified noise model the MPS-VQE noisy energy is -2.1490 Hartree, within 0.017 Hartree of FCI, whereas the hardware-efficient, qubit-UCC, SE, and UCCSD baselines fall to -1.6726, -0.6916, -1.5781, and -0.5293 Hartree respectively. The paper interprets this as evidence that the shallow MPS circuit limits accumulated gate noise, pre-training removes initialization fluctuations, and the neural-network extrapolation removes much of the remaining error.

Load-bearing premise

The entire accuracy claim rests on the assumption that a small three-layer neural network, trained on expectation values measured at a few artificially boosted noise levels, can reliably extrapolate the noiseless expectation value, yet the paper specifies neither the noise-scaling factors nor the training details and gives no comparison with standard polynomial extrapolation.

Editorial extensions

If this is right

If the claim holds, shallow MPS-structured circuits can substitute for much deeper chemistry ansatze in noisy settings, reducing both gate count and error accumulation.
Classical MPS pre-training gives a deterministic, physics-informed starting point for VQE optimization, which should reduce run-to-run variance and speed convergence.
Neural-network zero-noise extrapolation can be added to existing VQE pipelines as a replacement for polynomial extrapolation, as long as enough noise-scaled expectation values are collected.
Grouping commuting Pauli terms cuts the number of measurement rounds, so the same accuracy should be reachable with fewer shots on real hardware.
The reported $\mathcal{O}(10^{-2})$ to $\mathcal{O}(10^{-1})$ error range suggests that near-term quantum chemistry on medium-noise devices is plausible for small molecules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the method's practical value depends on how many noise-scaling points the neural extrapolator needs; a direct comparison against polynomial ZNE on the same folded circuits would isolate whether the neural fit or the compact circuit is doing the heavy lifting.
Inference: the same MPS-to-circuit construction should extend to longer molecules and to spin-chain Hamiltonians, and the pre-training benefit should grow with the bond dimension of the classical MPS, but neither extension is tested here.
Inference: the reported noisy energy of -2.1490 Hartree still lies 0.0119 Hartree above the noiseless MPS-VQE value, so the extrapolation does not fully remove noise; reporting median and worst-case energies across the 30 runs, rather than only the best, would give a sharper picture of typical performance.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper combines three known ingredients—MPS-style circuit pre-training, zero-noise extrapolation, and Pauli grouping—into one VQE pipeline and tests it on H4 under a defined noise model. What is genuinely new is the specific combination and the use of a small neural network for the ZNE fit. The code is available, and the gate count for the MPS-VQE circuit is much lower than UCCSD, which is a practical advantage. The pre-training results in Figure 2 do show a tighter energy distribution over many runs, so that part earns its keep.

The soft spots are real, and the stress-test note points at the load-bearing one. The headline noisy energy of -2.1490 Hartree comes from the neural-network ZNE described only as a three-layer fully connected network. There is no information on the noise-scaling factors, training data, hyperparameters, or validation, and no comparison to standard polynomial extrapolation. With a flexible fit to a small number of noisy points, that number could easily be a fitting artifact rather than a mitigated estimate. That concern is not hypothetical; it is exactly what the paper needs to rule out.

The secondary issue is also real: in Table II, the noiseless HE-VQE baseline (-2.1723) sits below the stated FCI benchmark (-2.1664). For a variational ground-state calculation this is impossible, which suggests either the reference energy is miscomputed or the HE-VQE calculation used a different Hamiltonian. Either way, it undermines confidence in the error comparisons. The best-of-30 reporting without error bars is a third problem, especially given the authors themselves note sensitivity to initialization.

These are not fatal flaws in the core idea. The MPS pre-training is legitimate, the circuit is shallow, and neural-network ZNE might work if specified properly. But as written, the central claim about noise robustness is not backed by sufficient evidence. A serious referee could extract a solid paper from this material, but the authors need to add full ZNE details, ablation against polynomial ZNE and no ZNE, error bars or distributions over the 30 runs, and a fix or explanation of the FCI inconsistency.

I would send this to peer review: it is a sensible, code-backed algorithmic study that deserves careful referee time. But I would not cite it in its current form, and I would not trust the -2.1490 number without verification.

Referee Report

5 major / 4 minor

Summary. The paper proposes a noise-mitigated variational quantum eigensolver, called MPS-VQE, that combines three ingredients: a shallow hardware-efficient circuit inspired by matrix product states, a classical pre-training step that initializes circuit parameters from an optimized MPS, and a zero-noise extrapolation procedure in which a three-layer neural network fits expectation values at different noise levels and extrapolates to the noiseless limit. The authors also introduce a commuting-grouping measurement strategy for Hamiltonian Pauli strings. Numerical simulations for a linear H4 molecule in the STO-3G basis under a specific depolarizing, thermal-relaxation, and bit-flip noise model are used to claim noise errors in the range O(10^-2) to O(10^-1), outperforming HE-VQE, Qubit UCC, SE Ansatz, and UCCSD baselines.

Significance. If fully substantiated, the paper would offer a useful contribution: a shallow MPS-inspired ansatz with classical MPS pre-training is a legitimate and potentially practical way to initialize VQE circuits, and the combination with ZNE and Pauli grouping addresses two acknowledged bottlenecks in near-term VQE. Strengths include the explicit noise model, the comparison of circuit metrics, and the provision of code repositories. However, the current evidence is only a single small molecule, and the central quantitative claim rests on an internally inconsistent benchmark and an under-specified neural-network extrapolation. The contribution is therefore promising but not yet established at the level claimed.

major comments (5)

[Table II] The noiseless HE-VQE entry (-2.1723 Hartree) lies below the FCI benchmark (-2.1664 Hartree). For any variational trial state, the noiseless energy must be at least the exact ground-state energy in the chosen basis; a value below FCI indicates an inconsistency in the Hamiltonian, the qubit mapping, or the FCI reference. This makes the entire comparison in Table II unreliable and calls into question the stated error ranges relative to the FCI value. The authors must correct the benchmark or the HE-VQE calculation and rerun the comparisons.
[Section II-F] The neural-network zero-noise extrapolation is described only as a three-layer fully connected network that fits expectation values at different noise levels and evaluates at zero noise. The manuscript omits the number and values of the noise-scaling factors (circuit-fold ratios), the amount of training data, the loss function, hyperparameters, regularization, and any validation against held-out noise levels. It also provides no comparison with standard polynomial ZNE or with no extrapolation. Because the headline noisy MPS-VQE energy (-2.1490 Hartree) is the extrapolated value, the central noise-mitigation claim is not yet supported. Please provide the full protocol and an ablation study.
[Section II-D / Eq. (7)] Pre-training initializes circuit parameters from an MPS, but the paper does not specify how the MPS tensors A[n] are converted into the rotation angles of the circuit in Eq. (7), nor does it demonstrate that the one-layer circuit can faithfully represent the optimized MPS for the bond dimension used. If the circuit ansatz is not capable of representing the pre-trained state, the benefit of pre-training is not guaranteed. The authors should describe the parameter mapping and report the fidelity or energy difference between the pre-trained MPS and the state prepared by the initialized circuit.
[Section III] All noisy and noiseless entries in Table II are reported as the best results from 30 independent experiments, with no mean, median, standard deviation, or number of shots. Best-of-30 reporting is sensitive to optimization and sampling noise and can create the appearance of robustness where none exists. The authors should report the distribution of outcomes, the number of shots per expectation value, and multiple noise seeds to support the reproducibility of the results.
[Section III] The experimental evidence consists of a single assumed H4 geometry in a minimal basis under one specific noise model. The conclusion that the method outperforms mainstream variational quantum eigensolvers is therefore overreaching. Additional molecules, different noise strengths, and a corrected HE-VQE baseline are needed before the comparative claim is supported.

minor comments (4)

[Section II-E] The text states that the Pauli Z gate has eigenvalues 0 and 1; the correct eigenvalues are +1 and -1. The intended statement is presumably about measuring computational-basis bits 0/1, but the wording is incorrect.
[Figure 2] The axes are not labeled in either panel. The histogram should specify the energy bins, and the lower panel should state what quantity is plotted as a function of what independent variable.
[Section III] The qubit encoding of the H4 Hamiltonian (e.g., Jordan-Wigner, parity, or Bravyi-Kitaev) is not stated, even though the encoding affects both the circuit structure and the noise sensitivity.
[Table I] It is unclear why the barrier count is reported as a meaningful metric and whether barriers affect the noise simulation; the authors should clarify this in the table caption or text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation chain is self-contained and benchmarked against external FCI.

full rationale

No circular step can be exhibited. The MPS pre-training minimizes the same Hamiltonian classically, but it is used only to initialize the quantum circuit and is not claimed as the source of the noise-mitigation result. The noiseless MPS-VQE energy is checked against the independent FCI benchmark, so the accuracy claim is anchored externally. The neural-network ZNE is a flexible extrapolation model fitted to expectation values at increased noise levels; it does not take the target noiseless energy or the final error as an input, so the reported noisy result is not equivalent by construction to a fitted parameter. Citations to prior MPS-VQE work and to MindSpore Quantum are ordinary references and are not used as load-bearing justification for the central claim. Concerns about the paper—underspecified NN-ZNE training details and the unphysical noiseless HE-VQE value of -2.1723 Hartree below the FCI benchmark -2.1664 Hartree—are correctness/reproducibility risks, not circularity. Therefore the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The reported performance depends on unstated choices (MPS bond dimension, ZNE noise-scaling schedule, neural network hyperparameters) and on assumptions that the shallow circuit can represent the pre-trained MPS and that neural extrapolation is reliable.

free parameters (3)

MPS bond dimension (χ)
The bond dimension used in MPS pre-training is not stated; it controls the expressiveness of the pre-trained state and thereby the quality of the circuit initialization, which directly affects the final energy.
ZNE noise-scaling factors (λ)
The specific noise-scaling factors and the number of data points used for the neural network extrapolation are not reported, making the noise-mitigation procedure irreproducible and its accuracy uncertain.
Neural network hyperparameters
The paper specifies only that three fully connected layers are used; layer widths, learning rate, epochs, and regularization are not given, and these choices affect the extrapolated zero-noise energy.

assumptions (3)

ad hoc to paper The one-layer MPS-inspired circuit in Eq. (7) can faithfully represent the pre-trained MPS wavefunction.
No mapping from MPS tensors to rotation angles is given, and a one-layer brick-wall ansatz with a single CNOT per site generally cannot represent an arbitrary MPS with bond dimension greater than 2.
domain assumption Noise-scaled expectation values form a smooth function that a three-layer neural network can extrapolate to zero noise.
Standard ZNE relies on smoothness of the expectation as a function of noise scaling, but extending this to an unregularized neural network is an additional assumption that is not justified or validated in the paper.
domain assumption The simulated noise model (depolarizing, thermal relaxation, bit-flip) is representative of real device noise.
The paper uses this noise model for all experiments but provides no hardware validation or comparison to measured device noise, so the reported noise tolerance may not transfer to actual hardware.

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Cite this review

Pith. "Pith review of Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation." pith.science (2026). https://pith.science/paper/TDD5ZKLG

@misc{pith2026250101646,
  author       = {Pith},
  title        = {Pith review of: Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TDD5ZKLG}},
  note         = {Machine review of arXiv:2501.01646}
}

abstract

As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the $H_4$ molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of $\mathcal{O}(10^{-2}) \sim \mathcal{O}(10^{-1})$, outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware. The updated code is available at https://gitee.com/mindspore/mindquantum/tree/research/paper_with_code/eigensolver_with_mps_and_ZNE or https://github.com/mindspore-lab/models/tree/master/research/arxiv_papers/MPS_eigensolver_with_ZNE.

Figures

Figures reproduced from arXiv: 2501.01646 by the authors.

**Figure 2.** Comparison of the ground state energy of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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