Title: Robust Integrated Learning and Pauli Noise Mitigation for Parametrized Quantum Circuits
Venue: NeurIPS 2025
Accepted: yes
Keywords: Quantum Machine Learning

Parameter Learning

Quantum Circuits

Noise Mitigation

Pauli Noise

Integrated Learning

Inverse Noise learning

Pauli-Lindblad equation
Abstract: We propose a novel gradient-based framework for learning parameterized quantum circuits (PQCs) in the presence of Pauli noise in gate operation. The key innovation in our framework is the simultaneous optimization of model parameters and learning of an inverse noise channel, specifically designed to mitigate Pauli noise. Our parametrized inverse noise model utilizes the Pauli-Lindblad equation and relies on the principle underlying the Probabilistic Error Cancellation (PEC) protocol to learn an effective and scalable mechanism for noise mitigation. In contrast to conventional approaches that apply predetermined inverse noise models during execution, our method systematically mitigates Pauli noise by dynamically updating the inverse noise parameters in conjunction with the model parameters, facilitating task-specific noise adaptation throughout the learning process. We employ proximal stochastic gradient descent (proximal SGD) to ensure that updates are bounded within a feasible range to ensure stability. This approach allows the model to converge efficiently to a stationary point, balancing the trade-off between noise mitigation and computational overhead, resulting in a highly adaptable quantum model that performs robustly in noisy quantum environments. Our framework is well-suited to near-term quantum devices in the noisy intermediate-scale quantum (NISQ) era, where noise is a significant challenge.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a gradient-based framework for training parameterized quantum circuits (PQCs) in the presence of Pauli noise, a dominant source of gate errors in noisy intermediate-scale quantum (NISQ) devices. The proposed method integrates noise mitigation directly into the training process by simultaneously learning PQC parameters and the parameters of an inverse noise model. The inverse model is based on a sparse Pauli-Lindblad representation, enabling efficient mitigation of Pauli noise via a dynamically updated, task-specific strategy inspired by the Probabilistic Error Cancellation (PEC) protocol. To estimate gradients of both PQC and inverse noise parameters avoiding the need for full noise tomography, the algorithm leverages stochastic subsampling and proximal stochastic gradient descent (proximal SGD) to ensure stable convergence while maintaining computational efficiency. Convergence and sample complexity bounds are established theoretically. The method is evaluated on a binary classification task using amplitude-encoded MNIST data. Comparisons are made with a noiseless PQC, a noisy PQC without mitigation, and a benchmark derived from Van Den Berg et al.'s pre-characterized noise model.\n\nSTRENGTHS AND WEAKNESSES:\nFrom my perspective, this work investigates the interesting topic of optimizing noisy parametrized quantum circuits. The assumption that the noises in the circuit are Pauli noise is practical in experiments. And the idea to combine error mitigation and parametrized circuit optimization can be interesting. However, I think this paper has a distance from the bar of NeurIPS 2025 due to the following major concerns. \n\nThe first weakness lies in the motivation for the work. There has been a series of works on optimizing quantum circuits in the presence of noise in a variational style. Some of these works focused on regarding noise as a part of the circuit and optimizing the parameters, and some others focus on searching for noise-resilient (parametrization of) quantum circuits. The authors fail to include these works in the literature review, and answer why we should focus on quantum error mitigation to correct (mitigate) the noise instead of the previous approaches. If possible, there could also be some numerical comparison between these frameworks.\n\nThe second weakness is the practical applicability of the proposed algorithm. In Eq. (5), the authors just directly optimize over all the parameters in Pauli noise and the parametrized gates directly. I am wondering if this simple construction leads to avoidable additional assumptions. As mentioned in the future works part, the framework assumes noiseless single-qubit operations, Hermitian linear observables with only $\\pm 1$ eigenvalue, absence of barren plateau, and omits SPAM (state preparation and measurement) noise. These assumptions should significantly restrict the usage of the algorithm on near-term devices that suffer from SPAM noise. In addition, assuming the absence of barren plateau would rule out the applicability of the algorithm to brickwork parametrized circuits unless the depth of the circuit is very shallow, the circuit is assumed to have some stringent structure, or the final observable is local.\n\nQUESTIONS:\nI hope the authors to address the major concerns raised in the Strengths And Weaknesses part.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-11"}

comments: SUMMARY:
This paper presents a novel gradient-based framework for training Parameterized Quantum Circuits on noisy intermediate-scale quantum devices under Pauli noise. The key innovation is the integration of learning both the model parameters and an inverse noise channel within a unified optimization loop. The authors propose an efficient gradient estimation technique to scale training under noisy conditions and validate their method on the MNIST dataset.

STRENGTHS AND WEAKNESSES:
Strengths:
•	Joint optimization of PQC and noise mitigation parameters is novel and well-justified.
•	Theoretical derivations and results are illustrated in a rigorous manner.
•	Experimental results demonstrate the proposed method’s strong performance.
Weaknesses:
•	Strong assumptions are applied, assuming idealized conditions such as noiseless Pauli operations and the absence of SPAM errors.
•	Experiments are limited to the MNIST dataset, potentially limiting the generalibility of the proposed method.
•	Missing ablations on the locality parameter c.

QUESTIONS:
1.	Please see the Weaknesses.
2.	Can the proposed method be applied to multi-class classification tasks?
3.	How sensitive may the method be to the listed assumptions? For example, if SPAM noise exists, how may it influence the derivation process and the performance?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nParameterized quantum circuits (PQCs) are a popular method in both machine learning and more general quantum computing for utilizing near and long term hardware in a way that is flexible and compatible with resources. A key challenge in using these models in the near term is the influence of noise on the resulting quality of the solution or results.  Previous methods for mitigating these errors have relied upon expensive or unstable extrapolations, or fitting of noise models that could later be approximately inverted.  This work addresses this problem in an integrated fashion, but integrating the noise corrections into the variational circuit itself and learning them alongside the parameters of the variational model.  In principle, this is not only conceptually simpler, but also more accurate due to the interplay between the variational flexibility in the parameters and the noise inversion process.  They test this method on a number of model problems and work out the theory in detail.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThe concept of integrating the noise mitigation into the overall variational circuit I feel is conceptually novel and interesting.  The integration of all the trainable parameters into a single framework has a way of simplifying how one understands the overall process and could develop further generalizations.  The performance the method gets appears comparable to existing methods while also being new.\n\nI feel this type of framework could be used to also generalize to non-unitary operations more generally being included inside PQCs for purposes other than error mitigation, though that is perhaps the topic of future work.\n\nWeaknesses\n\nWhile the method is conceptually interesting, perhaps the greatest weakness is the modest performance increases for the relative implementation and theory complexity increases.  Having non-physical operations within a parameterized circuit can run the risk of optimizing towards unphysical or non-realizable solution, running the variational nature of certain calculations.  This is addressed in this work by the objective function component \"G\" but a more clear discussion of this type of limitation would make the paper easier to read and understand how general the approach actually is.\n\nQUESTIONS:\n1. Can you clarify under what situations its possible to enforce that the inverse operations do not violate variational principles such that they can be used in a general setting without worry?  Is this possible to efficiently do in all cases? What are the limits?  A clear articulation of this in the paper would help boost the significance and clarity of the work\n\n2. It is stated that the operations that are inserted are not physical in the most general case, as they sometimes involve sampling operations etc.  However I do believe there is a subset of these that could be made physical using ancilla qubits and/or classical communication to strengthen the variational ansatz as well as mitigate noise.  Could you elaborate on when this might be possible vs not and what the significance could be more broadly?\n\n3. In a fault tolerant setting where we might assume local, random noise is quite small - how do you envision this framework enhancing the power of your circuits? How might the training approach be different?\n\nLIMITATIONS:\nI feel the paper would be more clear if the authors addressed the tension between variational freedom in the non-physical parameters and general variational principle violations that could occur due to sampling from effectively non-real states.  This would help understand the applicability and potential limitations for more general applications.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a method to train parameterized quantum circuits (PQCs) in a way that accounts for Pauli noise that occurs during gate operations. The authors propose jointly learning an inverse noise model(constructed using a\nsparse Lindblad-based formulation) alongside the usual PQC parameters so that training can adapt to the noise in a more effective way as compared to existing noise mitigation strategies like ZNE and PEC.  \n- While gradients for the PQC parameters can be computed efficiently using the parameter shift rule, estimating gradients for the inverse noise parameters is more difficult because it scales exponentially with the number of qubits. To tackle this, the paper presents a new gradient estimation technique designed specifically for these noise parameters, which helps make the overall optimization more scalable and robust. \n\n- The work provides comprehensive derivations of the gradient estimators, analysis of sample complexity and convergence, and numerical experiments showing how the method performs in practice.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The paper addresses an important practical challenge in quantum machine learning: mitigating Pauli noise during PQC training. It proposes a joint optimization framework that integrates learning an inverse noise model with PQC parameter tuning, which is a thoughtful and promising approach. This approach mitigates the need to apply predetermined noise models, making it scalable.\n\n- It introduces a novel gradient estimation method for the inverse noise parameters, tackling the scalability issues that arise as qubit counts grow. Clear derivations of gradients and explainations of how they can be efficiently estimated are provided.\n\n- Includes theoretical analyses of sample complexity and convergence.\n\n- Demonstrates the approach empirically with numerical experiments that illustrate its potential advantages over the relevant baseline.\n\n- The paper is generally well-organized and clearly structured.\n\n\nWeaknesses:\n- A major weakness is the making a large number of assumptions(see Section 3.1.2) to make the problem tractable\n\nQUESTIONS:\n- Of the assumptions you list, which do you think will pose the biggest challenge for applying your method to real hardware?\n- Do you see a path to testing this framework on current NISQ hardware, beyond classical simulation? What specific obstacles would you expect to encounter in those experiments?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-06"}