Robust Integrated Learning and Pauli Noise Mitigation for Parametrized Quantum Circuits

We thank the reviewer for the thoughtful critique.

Responses to specific questions:

The 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.

While the strategies proposed in prior efforts have demonstrated potential, they suffer from the following key limitations.

First, the ansätze employed in many of these methods are typically shallow, which limits their expressive power. Our work aims to address this limitation directly. As noise propagates through the circuit, wide and deep ansätze often give rise to noise-induced barren plateaus in the optimization landscape. This phenomenon significantly increases the computational cost of training, often rendering such efforts practically infeasible. Consequently, effective noise mitigation strategies are essential for scaling to deeper circuit architectures.

Second, prior methods often rely on the assumption of a static noise model to ensure the stability of training and model performance. For instance, noise-aware circuit learning approaches such as [1] optimize quantum circuits under fixed, tomographically characterized noise models. As a result, any drift in the underlying hardware noise necessitates re-tomography and full re-optimization. Similarly, noise-resilient parameterization methods like [2] demonstrate that optimal parameters remain unchanged under stationary Pauli channels, but do not correct for the bias introduced by noise.

Our method addresses this limitation by allowing dynamic recalibration. Specifically, when the structure of the noise model changes, our framework enables efficient adaptation without requiring full retraining. Instead, training can resume from the current parameter state, providing a more robust and practical solution in real-world settings, where noise models often vary over time. This capability is empirically demonstrated in Figure 3b, where our method outperforms static mitigation baseline despite only a small subset of CNOT gates being perturbed under a dynamic noise model—suggesting its potential to handle larger or more frequent noise drifts in realistic scenarios.

The 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 eigenvalues, 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.

Regarding other noise sources, such as SPAM (State Preparation and Measurement) errors, we acknowledge their impact on overall performance. However, our focus in this work is on gate-level noise. SPAM errors have been extensively studied in the literature, with numerous effective mitigation techniques available---see, for example, [3, 4]. These methods are often implemented in pre-processing or post-processing stages, and therefore, for the purposes of this study, we assume the absence of SPAM errors.

With respect to barren plateaus, all gradient based optimizers suffer from this problem. Specifically, barren plateaus are characterized by the conditions

$$\mathbb{E}[\partial\_\theta C] = 0 \quad \text{and} \quad \text{Var}[\partial\_\theta C] \leq \mathcal{O}(1/b^n)$$

for some $b \geq 2$, $n \geq 1$, where $C$ is the objective function. These conditions imply that an exponential number of measurements is required to estimate gradients unless the initialization is extremely close to a (local or global) minimum---a scenario that is both rare and difficult to identify in practice.

Finally, we would like to emphasize that the challenges mentioned at the end of the critique are inherent to any method that attempts to train variational or parameterized quantum circuits. Indeed, training such circuits is generally NP-hard, as shown in [5]. Our work aims to make this problem more tractable under realistic noise assumptions by focusing on scalable gradient estimation and mitigation of noise effects.

References:

[1] Cincio, L., Rudinger, K., Sarovar, M., & Coles, P. J. (2021). Machine learning of noise-resilient quantum circuits. PRX Quantum, 2(1), 010324.

[2] Sharma, K., Khatri, S., Cerezo, M., & Coles, P. J. (2020). Noise resilience of variational quantum compiling. New Journal of Physics, 22(4), 043006.

[3] Bravyi, S., Sheldon, S., Kandala, A., Mckay, D. C., & Gambetta, J. M. (2021). Mitigating measurement errors in multiqubit experiments. Physical Review A, 103(4), 042605.

[4] Van Den Berg, E., Minev, Z. K., & Temme, K. (2022). Model-free readout-error mitigation for quantum expectation values. Physical Review A, 105(3), 032620.

[5] Bittel, L., & Kliesch, M. (2021). Training variational quantum algorithms is NP-hard. Physical review letters, 127(12), 120502.

Many thanks for the thoughtful and insightful comments.

Responses to specific questions:

Q1: Addressing Weaknesses

Strong assumptions are applied, assuming idealized conditions such as noiseless Pauli operations and the absence of SPAM errors.

Regarding noiseless Pauli operations, we make this assumption based on the fact that Pauli gates generally exhibit lower error rates among one-qubit gates on modern NISQ hardware. Moreover, strategies such as Pauli Check Sandwiching [3] have been developed to further reduce their associated noise. That said, we recognize that Pauli gates are not fault-tolerant in the rigorous sense and that their noise can still influence overall performance. These second order considerations are addressed in subsequent efforts.

Other types of noise, such as SPAM (State Preparation and Measurement) errors, can impact overall performance. However, SPAM errors have been studied in literature, and effective mitigation techniques have been proposed [1, 2]. These strategies are typically applied during pre-processing or post-processing stages, and are orthogonal and complementary to our method.

With regard to barren plateaus, we note that all gradient based optimizers suffer from this problem, which limits the scale and scope of variational methods. However, even within this limited scope, faults pose a major challenge -- which our work addresses and provides important solutions for. We note that our solutions are independent of the barren plateaus problem and will be highly relevant even when the problem is resolved.

Experiments are limited to the MNIST dataset, potentially limiting the generalibility of the proposed method.

We acknowledge the limitation of focusing solely on MNIST, and we plan to evaluate our approach on more complex datasets in the final manuscript.

Missing ablations on the locality parameter c.

We note that our theoretical results hold for any positive integer $c \leq n$. In our work, we specifically chose $c = 2$ for the following reasons:

i) Any $n$-qubit unitary can be constructed from a quantum circuit composed of CNOT and single-qubit gates, which are universal. Thus, $c = 2$ is general enough to represent arbitrary ansätze.

ii) Two-local gates tend to experience lower noise, which makes the optimization problem more tractable.

iii) Two-local gates are more practical to implement on current NISQ devices.

Q2: Can the proposed method be applied to multi-class classification tasks?

One can generalize the measurement scheme to a positive operator-valued measure (POVM). In this case, the objective function has to be modified so that the model outputs the correct label with high probability. However, implementing POVMs is generally more difficult both in terms of design and hardware realization.

Q3. 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?

As mentioned earlier, SPAM errors can be mitigated in pre-processing or post-processing stages. The resulting performance under such errors thus depends on the quality of the chosen mitigation strategy. As for the other assumptions, relaxing the $c$-locality or barren plateau assumptions does not affect the mathematical derivations themselves, since they do not rely on these assumptions. However, the time complexity of the algorithm could increase substantially, and lifting the assumption of a constrained parameter space could result in an objective function that is no longer Lipschitz continuous, in which case the convergence guarantees no longer hold.

We thank the reviewer again for the constructive feedback and the opportunity to clarify these points.

References:

[1] Bravyi, S., Sheldon, S., Kandala, A., Mckay, D. C., & Gambetta, J. M. (2021). Mitigating measurement errors in multiqubit experiments. Physical Review A, 103(4), 042605.

[2] Van Den Berg, E., Minev, Z. K., & Temme, K. (2022). Model-free readout-error mitigation for quantum expectation values. Physical Review A, 105(3), 032620.

[3] Gonzales, A., Shaydulin, R., Saleem, Z. H., & Suchara, M. (2023). Quantum error mitigation by Pauli check sandwiching. Scientific Reports, 13(1), 2122.

Dear Reviewer,

Thank you again for your review and insightful feedback. We hope our rebuttal addressed your main concerns. If any clarification is still needed, or if you'd like to discuss any points further, we would be very happy to engage.

We sincerely appreciate your time during this discussion phase.

Best Regards.

Many thanks for the thoughtful and insightful comments.

With respect to the observed modest performance improvement, this is indeed the case for static noise models. However, the key benefit of our approach is in dynamic noise models. In this case, the improvements from our method are substantial. In particular, in Figure 3b, only three CNOT gates were perturbed under a dynamic noise model, yet our method outperformed the static mitigation baseline—suggesting that larger or more frequent dynamic noise drifts could yield even greater improvements. Notably, one of the key take-aways from our experiments is that our approach, while targeted to dynamic settings, yields slightly better performance even in the static setting.

There are two main reasons for introducing the constraint component $\mathcal{G}$. First, it restricts the search space to a convex feasible set and ensures that the parameters used in the inverse or recovery operation do not diverge. Second, it guarantees that the objective function is Lipschitz continuous over the domain. Without this constraint (i.e., removing $\mathcal{G}$), the objective function becomes non-globally Lipschitz continuous due to the presence of an exponential normalizing factor. This lack of global Lipschitz continuity implies that theoretical guarantees for convergence may no longer hold. Nevertheless, we observe that the objective function remains locally Lipschitz continuous. It is possible to generalize our convergence results to any convex and compact (i.e., closed and bounded) parameter space.

Responses to specific questions:

Q1: 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.

In order for the inverse channel $\Lambda^{-1}$ not to violate variational principles for all density operators $\rho$, $\Lambda^{-1}(\Lambda(\rho))$ must also be a valid density operator. In our work, we define:

$$\Lambda^{-1}(\rho) = q \rho - (1 - q) P \rho P,$$

and then

$$\mathrm{tr}(\Lambda^{-1}(\rho)) = q \, \mathrm{tr}(\rho) - (1 - q) \, \mathrm{tr}(P \rho P) = 2q - 1,$$

which equals 1 only if $q = 1$. Therefore, in general, $\Lambda^{-1}$ is not trace-preserving unless it is the identity map, which implies that no non-trivial $\Lambda^{-1}$ will satisfy variational principles for all $\rho$.

However, the composite map $\Lambda_l^{-1} \circ \Lambda_l$ can be physical under specific conditions. Using the definitions of $\Lambda_l$ and $\Lambda_l^{-1}$ (Equations 2 and 3 in our paper), together with the commuting property (Proposition A.10), we can rewrite the composite as:

$$\Lambda_l^{-1} \circ \Lambda_l = \bigcirc\_{k \in \mathcal{K}\_l} \left( \frac{1 + e^{2(\sigma\_{l,k} - \lambda\_{l,k})}}{2} I \cdot I + \frac{1 - e^{2(\sigma\_{l,k} - \lambda\_{l,k})}}{2} P\_k \cdot P\_k \right),$$

where $\lambda_{l,k}$ are noise parameters and $\sigma_{l,k}$ are inverse noise parameters. It follows that $\Lambda_l^{-1} \circ \Lambda_l$ is implicitly a valid quantum channel if $\sigma_{l,k} \leq \lambda_{l,k}$ for all $l \in [L]$, $k \in \mathcal{K}\_l$. Unfortunately, verifying this inequality without prior knowledge of $\lambda_{l,k}$ may require exponential effort.

We are happy to add this discussion to the manuscript.

Q2: 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?

The major difficulty in implementing $\Lambda^{-1}$ on quantum hardware lies in designing a quantum circuit $U$ that simulates the behavior of the non-physical term $-P \cdot P$, where $P$ is a Pauli operator. Specifically, such a circuit would need to satisfy the condition $U \equiv iP$ and simultaneously $U^\dagger \equiv iP$. However, this leads to a contradiction: since $iP$ is skew-Hermitian, we have $(iP)^\dagger = -iP$. Therefore, $U$ cannot be equivalent to $iP$ and Hermitian/unitary at the same time. As a result, we believe that there is no standard quantum circuit capable of directly implementing such an operation.

We specifically chose $\Lambda^{-1}$ to be non-physical because it is the only general inverse/recovery map for Pauli-Lindbladian noise. Suppose we attempt to construct a quantum channel $\mathcal{E} = q I \cdot I + (1 - q) P \cdot P$ that inverts $\Lambda = w I \cdot I + (1 - w) P \cdot P$. Then their composition is given by:

$$\mathcal{E} \circ \Lambda = (1 - q - w + 2 w q) I \cdot I + (q + w - 2 w q) P \cdot P.$$

For this composition to equal the identity channel, we must have:

$$q + w - 2 w q = 0.$$

This equation has only two solutions: $q = w = 0$ or $q = w = 1$. Thus, $\Lambda$ must be either $I \cdot I$ or $P \cdot P$---not a convex combination---to admit an inverse of the form $\mathcal{E}$. Therefore, we conclude that no physical (CPTP) inverse $\Lambda^{-1}$ exists in general for Pauli noise.

Q3: 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?

While fault-tolerant settings reduce error rates significantly, noise propagation still occurs. As noise accumulates across wide and deep circuits, the resulting optimization landscape can exhibit noise-induced barren plateaus, characterized by vanishing gradients and exponential sample complexity. This makes the training of such ansätz computationally expensive or infeasible. Consequently, effective noise mitigation remains a critical requirement for scaling variational models to deeper architectures, even in near fault-tolerant regimes.

We sincerely thank the reviewer for the thoughtful and insightful comments.

The reviewer makes two critical observations. With respect to other sources of noise, in particular, SPAM noise, we note that effective mitigation techniques have been proposed in prior work [1, 2]. These strategies can often be applied during pre-processing or post-processing stages, and are orthogonal and complementary to our core contribution.

With regard to barren plateaus, we note that all gradient based optimizers suffer from this problem, which limits the scale and scope of variational methods. However, even within this limited scope, gate-level noise pose a major challenge -- which our work addresses and provides important solutions for. We note that our solutions are independent of the barren plateaus problem and will be highly relevant even when the problem is resolved.

Responses to specific questions:

Q1: Of the assumptions you list, which do you think will pose the biggest challenge for applying your method to real hardware?

The assumption on single-qubit noiseless Pauli operators ( X, Y, Z) poses one of the most significant challenges for real quantum hardware. While Pauli gates in NISQ devices generally have low noise, the inverses or recovery operations are also susceptible to noise. These small errors can propagate, particularly in deeper and wider circuits, compromising the effectiveness of noise mitigation. Techniques to reduce this impact, such as Pauli Check Sandwiching [3], play an important role and are being investigated in our current work.

Q2: 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?

We are actively exploring adaptations of our framework to NISQ systems. In this context, a key challenge is the communication overhead between the classical and quantum hardware of the variational framework. Due to the frequent interactions between classical optimization procedures and quantum evaluations, communication bottlenecks affect runtime performance. Addressing this bottleneck is crucial to realizing a practical implementation.

References:

[1] Bravyi, S., Sheldon, S., Kandala, A., Mckay, D. C., & Gambetta, J. M. (2021). Mitigating measurement errors in multiqubit experiments. Physical Review A, 103(4), 042605.

[2] Van Den Berg, E., Minev, Z. K., & Temme, K. (2022). Model-free readout-error mitigation for quantum expectation values. Physical Review A, 105(3), 032620.

[3] Gonzales, A., Shaydulin, R., Saleem, Z. H., & Suchara, M. (2023). Quantum error mitigation by Pauli check sandwiching. Scientific Reports, 13(1), 2122.

Dear Reviewer,

Thank you again for your review and insightful feedback. We hope our rebuttal addressed your main concerns. If any clarification is still needed, or if you'd like to discuss any points further, we would be very happy to engage.

We sincerely appreciate your time during this discussion phase.

Best Regards.