Robust Integrated Learning and Pauli Noise Mitigation for Parametrized Quantum Circuits
Authors: Md Mobasshir Arshed Naved
Wenbo Xie
Wojciech Szpankowski
Ananth Grama
Venue: NeurIPS 2025 poster ()
Keywords: Quantum Machine Learning
Parameter Learning
Quantum Circuits
Noise Mitigation
Pauli Noise
Integrated Learning
Inverse Noise learning
Pauli-Lindblad equation
Primary area: other

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.

Robust Integrated Learning and Pauli Noise Mitigation for Parametrized Quantum Circuits
Md Mobasshir Arshed Naved Department of Computer Science Purdue University West Lafayette, USA naved@purdue edu
Wenbo Xie Department of Computer Science
Purdue University West Lafayette, USA x rie4010purdue... edu
Wojeiech Szpankowski Department of Computer Science
Ananth Grama
Department of Computer Science
Purdue University West Lafayette, USA szpan@purdue edu
Purdue University
West Lafayette, USA aygêcs purdue edu
Abstract
We propose 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 learn- ing of an inverse noise channel, specifically designed mitigate Pauli noise. Our parametrized inverse noise model utilizes the Pauli-Lindblad equation relies the principle underlying the Probabilistic Error Cancellation (PEC) protocol to learn an effective and scalable mechanism for noise mitigation. In contrast conventional approaches that apply predetermined inverse noise models during ecution, our method systematically mitigates Pauli noise by dynamically updating the inverse noise parameters in conjunction with the model parameters, facilitat- ing task-specific noise adaptation throughout the learning process. We employ proximal stochastic gradient descent (proximal SGD) ensure that updates bounded within a feasible range to ensure stability. This approach allows the model converge efficiently mitigation and computational overhead, resulting in a highly adaptable quantum model that performs robustly in noisy quantum environments.
stationary point, balancing the trade-off between noise
Introduction
Quantum systems offer the potential solve computationally hard problems in simulation, optimiza- tion, and machine learning. However, they pose significant challenges on NISQ devices [22]. The presence of quantum noise, which leads errors in state preparation, gate operation, and measure- ment [12 20). severely constrains the performance of variational quantum algorithms (VQAs) and quantum machine learning (QML) tasks. A major contributor quantum noise in NISQ devices is gate noise, commonly modeled as Pauli noise 29 Pauli noise occurs when unintended Pauli operators (X, Y. Z) are applied to quantum states during gate operations. These disruptions degrade the fidelity of quantum circuits and, in turn, the performance of parameterized quantum circuits (PQCs) commonly used in optimization and machine learning. Effective noise mitigation techniques are necessary to realize the potential of quantum computations on NISQ devices.
Pauli noise mitigation strategies, such cancellation (PEC), have several practical drawbacks. ZNE mitigates Pauli noise by purposely
noise extrapolation (ZNE) and probabilistic error
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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amplifying it, either by stretching gate durations or by adding noisy operations [T]] It then estimates zero-noise results through carefully designed cancellation strategies. This approach relies on precise calibration based on the hardware's 's noise characteristics. To find the zero-noise result, 7NE runs circuits repeatedly at different noise levels, which introduces overhead for complex cireuits or highly stochastic noise profiles. PEC utilizes detailed noise characterization, often by full or partial tomography to construct inverse noise operations [26]. For this reason, it is hard to implement for large systems [5] and highly variable noise profiles.
Van Den Berg's Sparse Pauli Lindblad Method 28 provides an efficient approach to model and mitigate Pauli noise by utilizing sparse representations of Lindblad operators. It puts forth principled framework for modeling quantum noise by integrating the dynamies of open quantum systems, This approach eliminates the need for resource- cesource-intensive full-noise tomography but relies on the careful selection of dominant noise terms, which requires domain expertise. It also presumes stable noise profiles and has limited adaptability to rapidly changing noise environments. This limits its applicability for NISQ hardware, where noise is unpredictable and dynamic. All of these methods aim to mitigate noise while being oblivious to the specific quantum task/ operator.
In this paper, we propose a novel gradient-based framework mitigate Pauli noise in PQCs addressing the challenges of adaptivity, efficiency, and efficacy. Unlike conventional approaches, our method rightly integrates the process of learning parameterized inverse noise models, constructed using sparse Lindblad-based formulation (28], with the optimization of PQCs in the training process. This simultancous learning approach allows our framework to adapt dynamically to task-specific objectives and noise characteristics. We present detailed derivations and theoretical foundations of our method, establishing its superiority, and demonstrate its performance in the context of real- world quantum machine learning scenarios. Our findings demonstrate that adaptive noise mitigation improves accuracy compared to state of the art baselines. Our integrated learning approach improves the reliability of quantum machine learning NISQ hardware.
Main Contributions: We propose a joint gradient-based framework that optimizes parameterized quantum circuits (PQCs) while mitigating Pauli noise arising during unitary gate operations, thereby enabling noise-aware training. As with all gradient-based approaches, efficient gradient computation is essential our method. While gradients w.r.t. PQC parameters can be efficiently estimated using the parameter shift rule [17), estimating gradients for inverse noise parameters is more challenging due to exponential scaling with the qubit count. To address this, we introduce a novel and efficient gradient estimation method tailored to inverse noise parameters enabling scalable and robust optimization of PQCs.
The remainder of the paper is organized as follows: Section 2 introduuess the necessary background. Section presents on gradient derivation while Section 3.2.2 describes our quantum algorithm to estimate gradients of all model parameters, including the inverse noise parameters. key component of our efficient joint optimization approach. Sample complexity and convergence results are discussed in Section 3..23 and Section [3.3] respectively. Numerical results are presented in Section 4 with related work and concluding remarks in Section 5 and Section
pressettss our proposed integrated framework for PQC optimization. Section 3.2.1 focuses
2 Preliminaries
2.1 Parameterized quantum Circuit (PQC)
PQC is a crucial element in the design of Variational Quantum Algorithms (VQAs). It represented parameterized unitary operator U-((0 (d) defined by:
( II ULb UL 1=1
(i(()) Vn
b =1
O.W.
where Ui(O)) exp Gi U,,,Y, (, 8n 0, are learnable parameters, Vi2 are non-Pauli constant unitaries, and 5 E (0. 1)a constant binary vector.
} is a parameterized quantum gate generated by Pauli string Gt E

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2.2
Noise Model
Pauli noise is a form of quantum noise frequently observed quantum computing and quantum information processing. It originates from stochastic errors in quantum gate operations, where each qubit in quantum system may experience noise process characterized by one of the Pauli operators. This type of noise can be described
probabilistic application of Paulis X, Y or Z to qubit, each occurring with a certain probability. Since any Markovian noise can be approximated using Pauli noise via Pauli twirling (8) the dominant noise affecting gate operations can be effectively modeled as Pauli noise. Based on this observation, we adopt the Pauli-Lindblad noise model introduced in 28 to define the noise model over noise-free quantum state p as:
(p) (ObEK: (wee/
w/. Ps. P&) (p)
(2)
where w3 = (1 + exp { 22 2 2 0.
Here, serves as placeholder indicating that the map is applied p. i... (() A(p). The notation OKer represents the composition of maps where each map corresponds to a noise model term in K. associated with a Pauli string Pk E {I.X, Y. )) @n and model coefficient Aç. These model terms reflect various noise interactions in the quantum system.
By definition, the non- non-physical inverse noise model to mitigate the aforementioned noise can be written as:
A (0) = (1 OMEK (q. - (1- Px P!)) (p)
(3)
where, Ok = (1 + exp (-20x))/2, =xp
Σ kEK
J o 2 0.
ok
Here, values of correspond to the coefficients of inverse noise model w.r.t. Paulis Pa. Remark 2.1. Throughout this paper, the notation for nested map composition ())) is defined as
Jt. o SL-1O.... 12 fi()) = [n( for any collection of maps f1, f fL.
2.3 Noise-Mitigated PQCs
noise-mitigated parameterized quantum circuit (nmPQC) an augmentation of the PQC, as described in Equation ( is carefully designed to minimize the adverse impact of noise. In contrast the conventional design where each layer contains only unitary transformations, each layer of the nmPQC consists of the gate U1 the noise model A₃ and its corresponding inverse noise model A, With this design, the noise Ar can be mitigated by the inverse noise operator A, thereby improving the performance of the PQC.
as illustrated in Figure [
Formally, an L-layer nmPQC, denoted by UR is defined as follows:
Un( A A Adu
where Adu (o) UL, U/,,, U2 is the 1-th gate unitary. Additionally, Ar and A, represent the noise and inverse noise models at gate I, as defined in Equation (2) and Equation (3). respectively, The noise model A; is parameterized by unknown noise parameters λt, while the inverse noise model A, is parameterized by learnable inverse noise parameters di, w.r.t Paulis p(!) associated with each unitary U1.
3 Integrated Learning Framework for amPQC
In this section, we propose a gradient-based framework to train parameterized quantum circuit on noisy quantum hardware, where the noise is modeled by Pauli-Lindblad model. We begin by formalizing the learning objective and outlining the assumptions underlying our analysis.
3

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Ui,bt
UL.b/
A
byyr PQC
layer Mitigtte PQC
Figure The diagram above illustrates single layer of the nmPQC. The complete nmPQC is constructed by concatenating L such blocks of this structure.
3.1 Problem Statement and Assumptions
Let (Pt, Mr) be elements of the data set D = {(r, Mr) E B(C*) x (-1,1)) where N = 2". We
assume that (pt, 9t) are independent and identically distributed (i.id.) samples drawn from a known we formulate the following
distribution. In order to train the model UR defined in Equation (4). optimization problem:
min
c(d, 0
(5)
J. 2,7 E [,,
(7, (.,
where the loss function L: RR
R is defined as
c 7 = DD
1
c, E 1 Σ (we -r (MMU
(6)
t=1
which follows a squared loss formulation and (M)RR(p)) denotes the predicted measurement
| 3) + IR U {+oc) enforces
outcome corresponding to the observable M. The function G
constraints on the learnable parameters by restricting d and to predefined domains. Specifically, Xsigma constrains the inverse noise parameters d for each gate, while Keheta bounds the variational
PQC parameters 7, ensuring they within a specified range and preserve parameter feasibility during optimization. The constraint sets are defined by Xzigma = -IT= I [0, B((.)), Ncheto =
I ( (1) 211), ((..k) Rzo, and (8( 2(1) satisfy -00 < 80 B <0.
Using these definitions, G( ) can be written
त, E Xsigma x Reheta 0 7) +o0 O.w.
In this framework, the learning task is comprised of: (i) optimizing the standard PQC parameters that define the unitary transformations; and (ii) learning the inverse noise parameters di,k, which mitigate the unknown noise effect of Ar. The PQCs are restricted to c-local Pauli operators, ensuring that each unitary UL,, acts on most qubits. As result, the number of inverse noise parameters σi, per unitary is constrained to at most 4", reflecting the overall number of Pauli strings in a c-qubit system.
To ensure feasibility and facilitate problem formulation, make the following assumptions:
Assumption 3.1. The noise parameters λt are constrained within the domain Xsigma, which ensures that each \ lies within the interval [0, 2 (t.kk)] C R>o-
Assumption 3.2. The single-qubit Paulis, I, X, Y. and Z, are considered noiseless.
Assumption 3.3. The State Preparation and Measurement (SPAM) noise is absent.
Assumption 3.4. M is Hermitian with eigenvalues -1.
3.2 Gradient and Estimation
As discussed in the previous section, obtaining the gradient of the objective function is essential for all gradient-based methods. this subsection, we derive the gradient of the objective function defined in Equation ( and present quantum algorithm for its estimation.

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3.2.1 Derivation of Gradient
First of all, we derive gradients of the per-sample loss function, (. Ø) (ye (MLLr((:)))
Using the parameter shift rule as described in 17]. we compute the partial derivative of Li ( 6)
w.r.t the PQC parameter B (from Appendix A.2.2 as:
( 0) ( (MMln(p)))) (Muls (A5 A, oUi (pt (of-")))- a0j
(8)
Here, we denote Uj = (ddes(0)+o) (0d for R, and introduce the intermediate result Pi (j-1) defined by: Pt 11-11= -OIE Adti,, (pt)-
Next, differentiating Le(d, δ) w.r.t. an inverse noise parameter ji yields (see Appendix A.2.
ac,(d, (v tr (MMlR(p))) tr Mihkss A7
Σ
(9)
gEGU
where = f G²- with p()) and the terms within the innermost parentheses arise from the Pauli decomposition of the Hermitian matrix Pt.j Ay Adu,,, (OE=AA, A Ado,.. (pt). Plugging in the identity
(1, p . = =0} is the subset of Pauli strings that anti-commute
stated in Proposition A.14 we simplify further to arrive at:
ac, t) -4 4y tr (MUUx(P))) (Miliss (A7 olp (p(s)))
(10)
where U ppo
[ - p( (p(
3.2.2 Universal Estimation Algorithm
By examining equations Equation 8 and Equation [0P. it becomes clear that the primary computational challenge in evaluating these partial derivatives arises from estimating the trace terms. These trace terms can categorized into three distinct types: tr (MMRR((P)) and (Mllsss (Adu(0)+a) (Ptz))) To facilitate the esti- mation of the relevant trace terms, we reduce each one to standard form and present the derivations in Equation (28). Equation (29). and Equation (30). Upon analysis of these expressions, we observe common structural form across them:
(Mll,sj (A7* Ad Adpta (pt.))):
tr (M AT Adw, At Adr;
(11)
where W; is Pauli string and Uj is unitary. Building this observation, introduce the universal estimation algorithm, described Algorithm 2 designed to unbiasedly approximate the expression in Equation ( (see Lemma A.16 for details). The corresponding quantum circuit implementation illustrated in Figure
Ado
Adw;
Ag
M
1-th layer of the algorithm, = 1,
Figure 2: (1) The quantum circuit implementing the universal estimation algorithm: operator W, corresponds to fixed Pauli string, while U/ are constant or parameterized (variational) c-local non-Pauli quantum operations. (2) Note that the inverse channels A, are non- -hyysicall and cannot be directly realized on quantum hardware. Consequently, the implementation of the circuit requires classical preprocessing. Further details are provided in Appendix A.3.I

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Importantly, this quantum-classical hybrid algorithm enables unbiased estimation of these trace terms on noisy quantum hardware without requiring explicit acquisition of the structure noise channels As, which is typically computationally expensive.
3.2.3 Sample Complexity
The universal estimation algorithm serves as the fundamental building block for the partial derivative
estimation procedures defined in Algorithm 3 and Algorithm corresponding to the partial derivatives
of L. given in Equation (8) and Equation (10p. essence, the universal estimation algorithm executed repeatedly to obtain unbiased approximations of the quantities defined in Equation and Equation (TO), up to a prescribed accuracy (see Proposition A.177 for details), Ultimately, by iteratively applying Algorithm and Algorithm
we obtain an approximation of the gradient of Cr
given point 7),
A natural question arises regarding sample complexity: how many measurements are required to observe that,
obtain reliable estimate of the gradient at (7 0 Referring to Lemma A.22 with probability at least 1 - 6, the number of measurements (or number of runs) required to estimate
(7.
a single partial derivative within an additive error & is no less than
32
In
2 ))))))
where Ia = exp (2 E/z EREK, ) This expression shows that the sample complexity (or the measurement complexity) can grow rapidly if E or is too small or the number of parameters (7, d becomes large, making scalability a significant challenge.
To address the scalability challenge, we incorporate two key techniques. First, in each update round, randomly select subset of coordinate directions update, where each direction is independently chosen with probability 1/p for some integer p 2 1 (see Algorithm This sampling strategy reduces the number of estimations required per iteration. The resulting sample complexity summarized in the following theorem:
Theorem 3.1 (Informal Version of Theorem A.23). Under Assumption 3.2 Assumption 3.4 The
expected sample complexity (measurement complexity) required for Algorithm to obtain a estimation of the gradient with additive ervor E >> 0 with probability at least where 8 E (0,1) 1) is at least
( B)
2 0.
2
1
322
In
P
2 ( )2
8p
Second, when the current parameters (d) are far from a stationary point, we relax the estimation accuracy to prioritize correct gradient directions over high precision. Concretely, we set the error tolerance to
r for some E (0. 1). In particular, by applying the bound established in
Theorem 3.1 the expected total number of measurements per update round at least .
322 2x
In
This adaptive strategy ensures that early-stage updates remain computationally
6p
efficient while preserving convergence behavior.
3.3
Optimization and Convergence
To train the model introduced in Equation 44. we consider the optimization problem defined in Equation (5) Specifically, we optimize the objective function in Equation (5) using the proximal stochastic gradient descent (proximal SGD) method, as described in Algorithm Unlike standard SGD, proximal SGD updates the parameters according to the rule
2 ((+1), ((+1)) + proxg (17(0, 7 (0) (0)) 7, C..)
(12)
where (1) 7, Lis denotes the estimated gradient of generated by Algorithm at the point ( 7 for randomly selected index i; [[D]], and proxg denotes the proximal operator associated with g.

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The proximal operator is defined as
proxg(ol1) ((t))
argmin (aleh)atch)) jo
(0(0(1),00 ,01)) +
1 221lo ((e) (olt) (olt)) )(1)))
where X' = Xsigma X Xtheta is the search space. This construction ensures that every element of generated by the update rule in Equation (12) remains within X, guaranteeing feasibility throughout optimization process.
the sequence { )
Algorithm 1 Learning Algorithm
Dataset: D ((pt:)e)
nmPQC (number of layers I., D' E 1R Measurement M, Total iterations T, Learning rate
Input:
schedule (1) - Number of shots Ñ (2Z) 20. and positive integer p.
(The non -on-iancaas learning rate schedule (1()) is such that r)()) (0, 1), and Et n/ is finite)
Initialization: V/ E for t = ,,..., do do & uniform(........ ()) (Sample a random index)
6 oo.
02 Uniform (3)) 2()); AI E [4], & E Kr 01 t 0.
p. ()emo
(4) Li, + ingradienteestimnee i,, M, 100, 7 yhi N.N.
[Algorithm 5
J,6 8
(++1), o (++1)) prox; (1710) (0)
(()) a. Li.)
end for
We now analyze the convergence of the learning algorithm defined in Algorithm I To begin, we present several key properties of the objective function and the estimated gradient. First, in Lemma A.28 we show that the function L is locally Lipschitz continuous over the compact set X, key property for ensuring convergence. Next, in Lemma A.29 we demonstrate that L G is lower bounded, thereby ensuring the existence of optimizer. Finally, in Proposition A.30 we prove that the estimated gradient produced by Algorithm (aatthmm]aaaaa satisfies the bounded variance condition, which is necessary requirement for bounded convergence behavior. Building on these properties, we summarize convergence rate of the learning algorithm the following theorem:
Theorem 3.2 (Informal Version of Theorem A.31 Under Assumption 3.1 for the non-convex objective function L. + G defined in Equation
(6) the convergence rate of the Algorithm 7 sstissies
El=1 Me Σ n [[(we)]] -[w) < p wo
3Lx Et
Th B max
P PI nEV
+
2 T/t
2 _
2 2 T/e
where w0 = (0) )(0)) is the initial point, T = (ō) T) denotes a stationary point, and
w, = (1),, (1) are the iterates generated by proximal SGD. The step size 17 E (0, 1) forms monotonically decreasing sequence such that Di 1)t < co. The constants V and Bmax defined in Theorem A.37
Remark 3.3. The factor in the bound above arises from the random selection strategy employed in Algorithm A larger value of p leads sparser estimated gradient ((() J. 2. fis, which in turn results in slower convergence rate.
4 Numerical Results
4.1 Experimental Setup and Evaluation
We evaluate our learning algorithm (Algorithm 1 on binary classification task on the standard MNIST dataset as in prior related efforts 3 30,, focusing on digits 3 and 6. The data is preprocessed via principal component analysis (PCA) reduce dimensionality, then normalized and encoded into
7

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R
R
Ry
Ry
Ry
R2
Rx
Ry
R2
Ry
Figure 3: A sample HEA layer features a 6-qubit Rx + Ry + R2 + CNOT design. Here, A and A- gate-specific; one-qubit noise and its inverse applied to Rx, Ry and Rz, while two-qubit noise models are used for the non-parameterized CNOT gates.
quantum states using amplitude encoding, with binary labels: -1 for digit 3 and for digit 6. A total of 5000 input-label pairs are generated, which are randomly shuffled and split into 80% for training and 20% for testing. To isolate training-time noise and mitigation, we directly initialize the simulator to the amplitude-encoded PCA state (i.e., no explicit cate-preparation circuit). For the nmPQC, we employ a noisy 6-qubit hardware-efficient ansatz (HEA), illustrated in Figure 3 consisting of two layers of parameterized single-qubit rotation gates Rx, Ry, and Rz, combined with circular entangling CNOT gates the base PQC. At the end of the circuit, all qubits We determine the initial learning rate, r7(1, via coarse-to- coenie-t search: wide log-spaced grid sweep to localize promising interval, followed by binary search within that interval to refine the value. During training. we maintain monotonically decreasing learning- rate schedule. The framework is simulated on a multi-core CPU computing cluster, and the simulation is implemented using Qiskit 10
measured in Z-
To assess the effectiveness of the proposed mitigation framework which augments the PQC with nmPQC (as in Figure B) we compare it with the following configurations: (1) A noiseless PQC, serving as best-case performance benchmark. (2) noisy PQC without mitigation, serving as baseline. (3) A mitigated PQC using a probabilistic quantum noise model, adapted from Van Den Berg al. (28]. We pre-estimate noise parameters using their procedure and fix them as inverse noise parameters during training. While their method is not originally intended for variational optimization, we adapt it as a benchmark, referring to as "Van Den Berg et al.'s mitigation approach". For training we use proximal SGD to optimize the model parameters along with the inverse noise parameter(for our method). Model performance evaluated by accuracy, while noise mitigation efficacy is assessed using mean squared error (MSE) over training epochs.
4.2 Experimental Results
3.9
Noie www.aaddnccccc..cc.........
521
saltooatonn ga
124
320
3.00
650
(a) Training loss comparison across noiseless, noisy and mitigated scenarios under static noise.
(b) Training loss comparison between mitigated scenarios under proposed mitigation method under dynamic noise.
(c) Convergence Behavior of our
probabilistic subsampling.
Figure 4: Training performance of the binary classification using MNIST dataset in the quantum noise context. Each epoch summarizes 50 iterations of the training process.
We evaluate the effectiveness of our proposed method by comparing its training loss and testing accuracy against alternative configurations the trivial scenario (i.e., full gradient estimation). Each experiment is repeated three times per setting, and the standard deviation is used to report variability. Figure 4a shows the convergence behavior of different methods over epochs under a static noise setting, where all CNOT gates in the noisy and mitigated models assigned fixed error rates. The

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noiseless model serves as best-case reference. As expected, the noiseless PQC achieves the lowest training loss. The noisy PQC without any mitigation mechanism exhibits significantly higher loss. highlighting the effects of quantum noise. Both mitigation strategies- our proposed method and that of Van Den Berg et al.[28] achieve an improved reduction in training loss compared to the noisy baseline. The Van Den Berg et al. approach pre-estimates noise models using the random benchmarking technique [Р] and applies their inverses using the learned noise parameters during training, yielding initial advantage. In contrast, our method learns the inverse noise parameters during training, eliminating the need for noise pre-characterization and calibration, and enabling streamlined, task-specific adaptive mitigation. We observe that Van Den Berg et al.'s approach achieves lower training loss initially, but as training continues, our method surpasses it, ultimately achieving slightly better overall performance. Table [ summarizes the experimental accuracies of different approaches, showing that our method outperforms the baseline noisy PQC and Van Den Berg et al.'s.
To evaluate adaptability, we simulate dynamic noise by slightly increasing error rates of three CNOT gates shortly after training begins for mitigated methods only. While Van Den Berg et al.'s 's approach continues using its static pre-estimated noise model, our method dynamically updates the inverse noise parameters during training. As shown in Figure 4b our method effectively learns the inverse noise parameters in response to dynamic changes and ultimately achieves lower MSE than the method of Van Den Berg et al.
Table Classification accuracies across Different methods
ACCURACY
NOISELESS PQC
94.13 0.31
NOISY PQC (BASELINE)
79.69 3.70
MITIGATED PQC (VAN DEN BERG ET.Al)
85.37 3.44
OURS
86.43 3.14
In addition, we investigate the trade-off between gradient estimation cost and optimization per- formance by evaluating the convergence behavior of our learning algorithm (Algorithm [ under probabilistic subsampling (see Appendix(A.33 (A.3.3). This experiment uses the same static noise profile as in Figure 44 In this setting, for a fixed parameter p, each coordinate direction (i.e., parameter) selected independently with probability p during each gradient update step. The case p 1 corresponds to full gradient estimation, while larger p values introduce sparsity in the gradient estimates.
Figure 4c depicts the convergence behavior for various p values over 50 epochs. For p 1, we observe faster and smoother convergence, with a steady decline in loss and minimal fluctuations, demonstrating the benefit of full gradient estimation. As p increases, convergence slows, and the loss curve becomes noisier with higher overall loss levels, reflecting the impact of sparser gradient updates.
2.00
220-
174
325-
Sen Bigeet
524
130
1.3
(a) Training loss comparison across noiseless, noisy and mitigated scenarios under static noise.
(b) Training loss comparison between mitigated scenarios under dynamic noise.
(c) Convergence Behavior of our proposed mitigation method under probabilistic subsampling.
Figure 5: Training performance of the binary classification using -NIST dataset (Pullover vs Shirt) in the quantum noise context. Each epoch summarizes 50 iterations of the training process.

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Additional results. We repeat the same protocol on the Fashion-MNIST dataset (Pullover vs. Shirt) [32; trends mirror MNIST under static (Figure Saj and dynamic (Figure (5b) noise, and probabiliisic subsampling behaves similarly (Figure 5c. Additional experimental details and the accuracy table appear Appendix [A.6.T] Furthermore, under the same dynamic-noise protocol across both datasets, we compare PEC and ZNE against our method (see Appendix (.66.)) in our
experiments, our framework adapts online and attains the best final metric (lower loss/MSE).
5 Related Work
PQCs are central to many quantum machine learning models, particularly in VQAs (2), originally introduced for ground state preparation [21]. VQAs adopt hybrid quantum-classien loop, where a quantum circuit evaluates
objective and classical optimizer adjusts parameters, This framework
supports applications such as quantum chemistry 15 combinatorial optimization 6 and quantum
neural networks (QNNs) 24 23 1]
Despite their flexibility, PQCs are susceptible to quantum noise, including decoherence, gate noise, and SPAM errors 222. Quantum Error Correction (QEC) offers robust protection [27], but its overhead remains prohibitive for NISQ devices. Instead, Quantum Error Mitigation (QEM) methods such ZNE 14 26 aim to reduce noise without heavy resource demands.
Pauli noise, common model of gate errors, can exponentially suppress gradients in VQAs, impeding training 8 31. Several approaches have emerged to address this. PEC techniques model noise as an invertible stochastic process, correcting via probabilistic sampling (26). Learning- based strategies such as those by Strikis et al. [25] and Czarnik et al. [4] train on auxiliary circuits to estimate mitigated expectations. Van den Berg et al. (8 propose incorporating Lindblad-modelee
noise directly into the optimization. A recent diffusion-inspired approach [19 mitigates noise via dynamics.
learned
6 Discussions
We propose gradient-based framework for jointly learning parameterized quantum circuits (PQCs) and inverse noise operators, enabling noise parameter estimation during training. This approach supports rapid recalibration when the noise model changes, thereby avoiding full retraining and improving adaptability on near-term devices. A parameter-shift- hardware-efficien gradient estimation with respect to inverse noise parameters, ensuring robustness under Pauli noise. While effective, the method faces challenges: parameter growth under low locality (e-g., e. a(log n)) can increase computational cost, and the reliance on simple measurement may limit broader applicability. Future work will focus on reducing the complexity of inverse noise operators and extending applicability general measurement schemes.
rule introduced for scalable,
Acknowledgments
This work is partially supported by the NSF Center for Science of Information (CSol) Grant CCF- 0939370, and also by NSF Grants CCF-2006440 and CCF-2211423.
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12

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A Appendix
In the following appendices. we provide additional technical details that complement the findings presented in the main manuscript. In Appendix(A. A.I we offer concise overview of key concepts in quantum computing and properties of the Pauli-Lindblad noise model. In Appendix A.2 we present detailed derivation of the gradient of the loss function. In Appendix A.3 we describe the algorithms used for gradient estimation. Appendix [A.4] we discuss the sample complexity of the quantum
algorithm. Finally, in Appendix [A.5] we characterize the convergence behavior of the proposed quantum learning approach and provide an analysis of performance.
A.1 Preliminary
A.1.1 Overview
We present a brief overview of some key concepts and definitions in quantum computing. Definition A.1 (Space Linear Operators), If H. be fiite-dimensional Hilbert space, then let B(H) be the set of all linear operators on H, that is:
( (A: H H}
Remark A.2. In this paper, we assume H C.N. where N = 2" and is the number of qubits in the system. In other words, B(C*) is the set of all linear operators from CN to C..
Definition A.3 (c-local). If Hamiltonian H is supported on at most e of the 2 c qubits (i.e., H acts on qubits in S S [n)]. where |5] = c), it is called a c-local Hamiltonian (on n qubits).
Remark A.4. this paper, we say the locality is high when c = O(log n)), where n the number of qubits.
Theorem A.5 (Quantum Channel (0)). A map & B(H) + (B++) is a quantum channel if and only if satisfies the following three conditions:
1. (Trace Preserving) tr (E(p)) = for all mixed states p on H.
2. (Convex) (E, P.P.). = E (p.), where {p:} is a probability distribution.
3. (Completely Positive) For all positive Hermitian operators A E BBH),, E(A) is also positive operator.
A.1.2 Pauli Matrices and Pauli Strings
The Pauli matrices L, X, , Z are defined as the following 2-by-2 traceless complex matrices:
I 9). x= 1) y= 0
: 1 ) z
0 -1)-
Pauli string P of length n is an element in {1, X, Sn Since 1 (Z)) X
1 (Z)) =i. (Z)1 and Z ( (Z)), any Pauli string P E (T, X, Y, 2) On
can
be uniquely expressed as follows:
P E X xX) 2
where & ti, 7.7 1)" are binary vectors, and マ() (Z()) represents the j-th component of the vector & (z). Given this observation, we can now define the symplectic inner product between two Pauli strings.
Definition A.6 (Symplectic Inner Product (Less General)). If = & (x)
and Pa E 0" (X)= 72(1) 2)=2(1) are any two Pauli strings, the symplectic inner product between Pt and Pz. denoted by (P1, P2).,p is defined by:
(P1, P2).p = (Z) Z2+
(2)
mod 2.
20

---

The following proposition is direct application of Definition A.6 Proposition A.7. Let P1 E 8 (x) and Pa E2 (1) 22() be any two Pauli strings. Then,
PiP (-1)
Sketch Proof of Proposition A.7 First, we note that for any b1 b2 {0, 1}.
(x)* (2) (X)*
Using the properties of Pauli matrices and Kronecker product, we obtain
P2
(X) 7 (000 (2) (
72 (
(2) 7z(1)
1
j=1
(X) 7 (1) (Z) 71 (x) x2(0) (2) 22(0)
j=1
(-1) 210)) #20) (x) (x)) z1() (2)
=
==1
[1) 3.0)
(-1)E3- 2,00
(X) x2 (2) (x) = (2)0))) 310)
j=l
-
X Bz (Z) (2))) 7 (2)) 2:00
-f1 (11 mod
(X) 22 (x)
P(- P2P1
The following corollary follows immediately from Proposition A.7 Corollary A.8. Two Puuli strings commute if and only if their symplectic inner product is 0.
A.1.3 Pauli-Lindblad Noise Model
As suggested in (28, the Pauli-Lindblad equation with no internal Hamiltonian dynamics can be written as:
at (( = Lp(t), where C(p) = Σ (p(PpPP pp)
(13)
REK
and the noise model derived from Equation can be expressed as the following quantum channel:
A(p) OKek (wp (1 - PkpP)
for some K. C [4" 1). where W/ (1 exp - 2A 1)/2. To construct the inverse channel, 228 used the probabilistic error cancellation (PEC) protocol. The PEC approach allows for the application of inverse noise channel to correct the errors introduced by the noise channel. The inverse channel typically constructed by taking probabilistic approach to undo the effects of the noise. The inverse noise model is by definition is non physical map and expressed
A 11 (p) Okek (HAP (1 wr.) PxPPPP
21

---

with y= exp {EkER 2 represents sampling overhead. For each & E Æ. the identity is sampled with probability Wi or the Pauli Pk is applied. The number of non-identity Paulis are counted and the final Paulis are computed as the product of sampled terms. This process is repeated for all layers and the corresponding number of non-identity Paulis recorded along with the sampling overhead. For layer = ... L, let the respective count be m, and overhead be "Y. then the final measurement outcome is multiplied by II (-1) get an unbiased expectation value average.
For our purpose, we reiterate some useful properties of the operator (al I 6P% P for any Pauli string Ps and a. b E R.
Proposition A.9. Let Pk be any Pauli string. Then, (al + bPg P is linear.
Sketch Proof of Proposition A.9 Let A, B E B(H) be any two linear operators. Let e, E IR. Then,
(al 1 bP: (e) + dB)
=a](cA + dB) + (c + d
=caAAI + cbP + dolBI dbP
=c(al + I+ I (A) + d( bP (B).
Proposition A.10. Let Pky Pk2 be any two Pauli strings. Let a, b, c, d E TR any real values. Then, a bP,, Pl and o dPx, Pl commute.
k.
Sketch Proof of Proposition A.10 The proof is basically the same the one used in 28 First, we write + bPr, Pr₁ and
dP Pk₃ in their matrix form as the follows:
al & 1 bPa, 2 (Pt)"
and
cl & 1 dP s (PL)
Then, by applying Proposition A.7
obtain
(wt x 1 bP,, & (PL )")
I I @ (Pt))
=ac(1 1)(1 1) ad(I 8 1)
Prr (P
+ (Pr, 0 =
@ (P!))) (Phs (PL))T
(T [t0
+be
(hh 20 (PL))) (Ph s (Ph)²)
=(JOT+ @ (P) (aT 1 bPr, @ (P))T))
Remark A.11. The composite map A-1 A can be physical under specific conditions. Using the definitions of A and A (defined Equation (2) and Equation 3), together with the commuting property (Proposition A. 10), we can rewrite the composite as:
- Oke[L,--.49) .15) (1+ (c)
2 de) Pe Pk
1.
2
where A are noise parameters and σr are inverse noise parameters. It follows that A A is implicitly valid quantum channel ax & A for all K. Unfortunately, verifying this inequality without prior knowledge of d may require exponential effort.
22

---

A.2 Gradient
Derivation of Str( (Br(M))(()))
A.2.1
Using the linearity of Un and operator ōøz.
a
we can show that
Ötr (MMUR(P)))
-IrMM ( = tr (M oue (AT((P))))
(14)
00j4
joj.
d0
where
Ptj = Ag
Adusis
(O o Advio)) (pc) and (e,z(()
(Of=H1AA7 Ar Adu:. (0).
Proposition A.12. Let C1, C22 C3, C4 IR be any constant real values. Let P be any complex matrix. Suppose IR CNNN is any function depending on the variable x E R. Then,
0
Ör ((1 + +2)
T 1
+ c4) Pt) (A(x))
=((c + 22) (ct Pt)
A(x))
Oz
Proof of Proposition A./2 Plugging in the matrix (xx) into ((1 + 02) 1 (cs +cs Pa) Pt) on the left-hand side, obtain:
((c: + 02) . + (cs + PP Pt) (A(z)) (cr + c2)A (ca (c)
Taking the derivative with respect to x, we get,
0 dr ((c) (2))((2) + (cs + cc))) = (cr + 62)
(AA(x)
(0A(z)
+ (c3 + ca)
cc)P
pt
0x
Or.
which is exactly the same as the expression on the right-hand side.
Using Proposition A.12 it is straightforward verify that operator U4ys commutes with d0, Therefore, Equation 14 simplifies to
M.es> (A, (pss)) = tr Mill,>s
0A, (AA
))
tr M
(15)
0osa
Next, applying the Pauli decomposition of Pajj. we obtain
Paj
Σ
Gj.99.
(16)
(EX,YXYY
Combining Equation
15
Equation
16 and Proposition
A.9 we arrive at
Our (MUnlpp)
(0)7
Σ
aiigtr
Mehiss
(17)
DE(I.XY.2J..
a 00,,a
To proceed, we need to compute the partial derivative, the following proposition.
First, let us simplify 1 (g) using
Proposition A.13. If P.g g are any two Pauli strings, then
(a + b)g
[P.g] =
(al 1 + 6P Pi) (g)
{
(a b)g o.w,
for any b E R.
23

---

Proof of Proposition A.13
By definition, we have
aaToT+a bP TP Pi) (g) a-g---0 b- PgPi
Considering two cases:
Case 1: If [P.g] = 0, then
a g + b. Pgpt -a-g bpp..pPPi--p. -a
. Case 2: [P..] # O, then by Proposition A.7 we know gP = (-1)Pg which implies:
a.g g a.gbb. b. PyP a.g. g- b.gppttppp.............pppp..b (a- - b)g.
Using Proposition A.13 we deduce, for all g E (1, Y, 2] 100 20
A (g) =y OKEK, (0 I + (gg, 1) (P) (g)
(g)
=
II
(2gj.k 1).
II
g
KE
kEK;
(p .9)
(p()) .9)
=
(2gj.t 1) g-
kEK; (p(2).o =0
Recall that 7 = exp { terms, we obtain
(2 EiEK,, Jjk and qio
1+expl 20
Substituting the corresponding
35
II
(2gj2k - 1)g
REK,
(p()) .9) =0
1 + exp (-20j,k) -220
exp
2 Σ Jji
II
2
2
K.K,s
keK, (p(2..) .n) =0
exp
Σ
JjK
exp -0 .g
kEK,
REK, (p(2) .0) =0
2 Σ Oj,k
II exp
-2TT,k I (P())))) up
= =
exp
kEK.
kER,
= = =0]} g.
2 Σ Ojik
2 Σ JJK II (p()) .)) sp
exp
REK:
REK,
24

---

Here, [[:] is the indicator function. By splitting 2 2 jj into 2Σ
keR,
j and
(pl 3) =0
2
kEK, (h () m
J and rearranging terms, we obtain
=1
A, (g) = cx
Σ
jik
jjb
cxp 2
Σ
0
KEK, (Pf .9)
kEK
(1(2 .9) op
(P( .9) jjk
= exp
9.
(18)
kEK,
Next, applying chain rule to compute the partial derivative of (g) w.r.t. Tjo. we get:
(g)
0A)'(9)
(g)
δ exp
Σ
Jik (PL2) (g
00 1.
jojn
KEK,
= exp
JiK
(pl2) .9)
2 p() . 9.
(19)
KEK,
Combining Equation
17 Equation
18). and Equation
we have
Ötr (Mll((p()
(20)
Σ
ajigt MMlliss exp Σ J bEK,
(pl)) (9)
2
p()
gE(1.XYY2200=
up
sp
Σ
op p(i) aj. 2
,9)
tr
MMlliss
exp
2 Σ
jjk
(D(.)) .9)
gE|L,XY,2100
kEK,
sp
Σ
20gght Mlhi,s
exp 2 Σ
Oj
(p()) ,g
99
VE(LXXYY,200 (1(2) (1(().
kEK,
sp
.)
Σ
20g (Mldsss (A
(E (p()) .2)
=2tr
Mlde,ss
A7
Σ
ajgg
(21)
0E(LXX,Y.2) (p(2), .9),
The expression presented in Equation the following observation.
21
impractical for direct use. To address this, we consider
Proposition A.14. If P E X1, 2z 8m is any Pauli string. then
Σ
OigD9 = (pr. Ppt.5 Pt)
1
2
vE(I.XYY22@. (P.g),-
25

---

Proof of Proposition A.14
Again, applying Proposition A.7
the linearity of the adjoint operator, we
obtain
Pt
Σ
Σ
j.gg.
(22)
0E(I.X.V.2). (P.9),p
gE|1.X.Y.2)0N (P
Combining Equation (16) and Equation
we have
(pts - Ppt.P) 2
Σ
Σ
Σ
ajgg
2
DE(1.X.Y,2)00
DE(L.X.Y.2))8 8m (P
9E(I.X,Y.21 on (p:
Σ
Σ
0j.99
2
gE(1.X,Y.2)38
DE(I,X,,,21 8m
(Pa)), by 00
(P.g) op
=1
Σ
Σ
2
(E,Y,,Y, Rn
(E(1,X,Y,2) (Pi
(P.a) =0
1
Σ
Σ
2.
0riig
2
sE(LXX,Y,20 (P.9),p
sE(LXXYY22) (P.9),p
By applying Proposition A.14 we are able reformulate Equation 21 into a form analogous the parameter-shift rule. Specifically, we obtain:
Otr (MM)(())) 2
1 2
J (Milsss (A5² (pts))) (MLL ((D)
(pllt Pe. (P(((())))]]
00
A.2.2 Derivation of (tr(Mi/g(e.))
00,
We include both parameterized gate and constant gates in the PQC using Equation (T). therefore, we have to make sure that gradient calculation is exclusively performed for parameterized gates, i.e., model parameters only. As a result, we derive the expression for @u(Miin(e)) for < j s L such 09,
that by = 1 to ensure gradient calculation for appropriate model parameters.
Using the definition and linearity property (Proposition A.12), follows:
we express the partial derivative as
Ou (MURR(P)) =tr
MUk A o A,
(Adu,,, (o(--1)
00,
00,
=tr
MUh,>
A A,
(Adu,(e)) (pl
00,
=tr
MUlsss
A A,
0
(2(0)))11110000
00,
26

---

where o (Of=lA, Ar Adoi... (pi). The partial derivative of ()()) -)
with
respect to 0j is given by
0U, (U,(0,) (U)())
002 Pi - [0)=0()))+ OU,
303
003
= -,,110000 (00
= -(C,,,,(,)pp='"')}}100
Applying the parameter-shift rule given in
we obtain:
(U,(0;)pp -
00;
TI (1, (0,+ PE - (0, (0+++))
1 (u, (0, T
1
D (0; -)))
2
Pt
Altogether, (u(MM(n(Ps)) is given by: 86, atr (MUxx(P))
00,
1 2
(Mlliiss (Aj1 o Ady (pe (
- (Miliis (Aj1 Adu, (18-11))
A.2.3 The Gradient of Loss Function
Given the observations above, applying the chain rule, we obtain:
i (yt- tr (MMUn(()))) -4 (y (MU(R(P))) ir (Mehiss (A7² oppe (pts))) (23)
where Uptn _/Ad, AAd and:
(yi- - tr (MMUR((P))) -2 (ye - -tr-tr(ee (MMUR()))) tr (Mihiss (N51 o A, oU (Dg-11))
00,
(24)
where Uz (dd [0, for E R.
A.3
Estimation of Gradient
A.3.1
Universal Estimation Algorithm
In this section, we introduce the universal estimation algorithm, quantum-ccassieal hybrid method designed to estimate the quantity
tr (MOOL AT Adw; o A o Adu;
(25)
Here, M denotes Hermitian observable, P is density operator, W; is a Pauli string, and U/ represents an arbitrary (non-Pauli) quantum operator. By construction, each composition Adw, A₁ o Ady, defines valid quantum channel, and can therefore be implemented directly quantum hardware. The primary challenge lies in the realization of the non-physical inverse channels A,
Fortunately, simple classical preprocessing step enables their indirect implementation. Recall that each A, is defined linear combination of identity and conjugation by Pauli operator, i.e.,
qe.t - (1 -11g).))))) p() kk where dr.i (0. 1). To realize A, we sample from the Bernoulli distribution Bernoulli(1 I..)).
. If the outcome is O, we apply the identity operation; the outcome
1. we apply the Pauli string p()) and record classical multiplier of 1. After measuring the quantum
27

---

state at the final stage of the algorithm, we multiply the final result by (-1)², where S is the number of such -1 recorded during the run. The detailed implementation presented in Algorithm 2
Algorithm Universal Estimation Algorithm (universal (uiierrsaleestimator))
1 E [[P]], Measurement M, Parameters d' E Xsigme: 7 Xiieta
Input:
Input: set of constant Pauli strings (W WL) [By assumption 3.2 all Wi are noiseless)
S 0
Prepare the quantum state Pt
for 1 = 1 I. do
(Apply the inverse channel AT
Pt 4 Adw, A₁ (p); (Evolve the Quantum State)
(Here, for Adg;, we look up the corresponding parameter 0 in 0 when Uj is parameterized.)
for Kc do
q1 1+exp(-20L.A)
bL. ~ Bernoulli(1 q(
2
if bI. then
P (Apply Identity)
else
Pt ( pepppppppppppp p(P())) (Apply (())
t S+1+
end if
end for
end for
Measure the quantum state p on M and let O be the outcome of the measurement.
(0) Return exp 2 (ZEe o(.)) (-1)5. O
Remark A.15. By construction, the universal estimation algorithm does not impose stringent require- ments on the locality of the quantum operations involved (see Remark A.4). However, when the locality is too low-for instance, when = n-the structure of the inverse channel A, can become significantly more complex. In such cases, the cardinality of the corresponding index set K,, may grow exponentially with the number of qubits n, rendering the algorithm computationally intractable.
We now proceed to show that Algorithm 2 yields an unbiased estimator of the quantity (M O AT Adw, o Ar o Adu((p)))
Lemma A.16. universal_ estimator (Algorithm P provides an unbiased estimation of ir (M Of A, Adw, A o Adv; (pt)). where M is Hermitian and Pt is density operator.
Proof of Lemma A.16 Let (bI,,)
10,1) be set of independent Bernoulli random variables 2
with parameter Qt.k 1+xxp(-2a,.) We denote by .(b,,)) the quantum state right before
the measurement M, which depends on the choice of by From the algorithm definition, we note that (-1)s and P(br.)) random variables depending b and they can be re-written follows:
(-1) (-1)E
and
P(b.,)) = O (OkEK, Ad Ad Adw, o A Ady,
(26)
Here, (d) A At is the adjoint map and the bullet stands as a placeholder. For clarity, we
re-express Equation 26 by vectorizing using the identity, vec (ABC) = (A @ V [B]. We then obtain:
p & (1=(9))))) bb Ni vec (pt,
vec [p(
II KEK.
28

---

where Mi is defined by:
II KEK,
(w:)") (149))7) (mo
(ULb. (U/mm)
Taking expectation over bi, we arrive at:
E
(- 1) Ei
ben ben
vee [pcewil]]
br
II
p @ )() mue v vee [ao]
E
(-1) bi,
II MERI
bi.
E
II MKEK
-P(()
(17())))
) Mi) vec [pi]
bi.k
M=L
(-P!)) 60 (1P!7)) )")
b1.k are also independent.
Since by, are independent, it follows that the terms Moreover, A and Pt do not depend on bi Therefore, applying properties of expected value, obtain:
(-1)Enubee bus vee [p(o..)]]
E
bi.k
(-PA") P1"0 2 ((1(()))²) be. 1) Ni vee [pel
E
KEK,
qu (-pA") 2 (PP())))))
[=L
AREK
+(1 qq,x) (-eA") x ((P
1) No) vee led
a4 18 1 - (1 q(x)) p!l & (1P9))))) Ni) vee [pa)
(27)
kEE
Re-writing Equation
27
we express the expectation as:
E b,k
(11)2ina bue PrOnNs]
O (OKEK, (mgAdr - -(1 q..x)dddp))
((lW))(()) )())) Adw; Ar Ado;) (pt) 1 1-m OKEKa (g..d, -11- Ad(p)) )) Adw; o A Adv. (pt)
Of
72
Of/
(177') o Adwi o d Adu. (pi).
(m)
where 71 exp (2 E
kEE, ). For a fixed bi,s, one can easily verify that 2) is a density operator. Using the property of quantum measurement, we note that the expected measurement outcome O is given by tr (Mp()))) Combining these results, the expected value of the output of
29

---

the universal estimation algorithm is given by:
Σ
(-1)s 0
exp
L,k
exp
2
Σ [E[1,Z]AEER
Iik
(t)
E (-1
(-1) Ein be, ②
exp
2
Σ |E[I.L],AEK;
(1)
E
E
(-1)20n bi, 0
b1,
exp
2
Σ |E[1,L],AEK:
i
(t)
E
(-1)Eut
bir bik (Mprocal)
b
cxp
2
Σ LE|E,
Oik
(0)
tr M
E
(-1) prblat))
bu.
(M
(Ot
(1arT))
Adw, Ar o Ades (ot)
n
711
1=L
M
Of
?1
12.-A7 A7
1
Adw; o As o Adv;) (pe)
7
=tr (MO
A
Adw, o A o Adu;
This algorithm serves as a foundational component in the estimation of partial derivatives, as discussed in the following section. There, we will show how the partial derivative estimation problem can be reduced to the trace estimation problem stated in Equation (25)
A.3.2 Estimation of Partial Derivative with the Universal Estimation Algorithm
To estimate the partial derivatives with respect to specific parameters, first require effi- cient estimations of the trace terms: tr (M))))))), tr (ML,,s) (A, o ad poo (Pis)) ). and
(MUl,>s (Adu(e,+e) (pt)))) To facilitate their estimation, first reformulate them. In partieu- lar, the term tr (M)))(P:) can be expressed
tr (MO AT o Ad, Ar Adu (pi)
(28)
the term tr (Mels>s (A7* o Ad (pr)))) as
ur (MM (MOffttiAA Ad, o A Adei.,) o (A5' o Ad og o Adds)s))
(29)
(OfEAAT o Ad, Ar Adrim) (p+)):
and the term tr (MLl, (Adu(05+a) (pts))) as
tr (M OOE o Ad, Ar Adv (pr)..
Ad, o A e Advi.. (A7 o Ad, A, o Adw;
(30)
Building on the transformations described above, we can efficiently estimate the three trace terms using the universal estimation algorithm presented Algorithm 2 estimate the partial derivatives defined in Equation 23 and Equation (24). integrating the observations discussed above, we introduce
30

---

two estimators: Ligma_grad_est and theta_grad_est, explicitly detailed in Algorithm and Algorithm respectively.
Algorithm 3 Estimation of 0(v. M(M (sigma_grad_est)
Input: [[[]] Measurement M, Parameters 7 E Xsigmas 0 (0) E Neheta true label yt j E [L], q E j,, and positive integers N N(ia),
O O ((200)))) < 0,0
for = to Ñ do
! f inivrrall_sttmmato (t.M, -. 8. (I, (I,....., .1) [Algorithm
0 0 + end for for N:(.9) do
Bernouli( if b = then
Oa (m.
11)
f- universal_estimnee
else
Oa (2.05.a) t (-1) universal_estimaan (p,. J, B. (I, (I.....11 P! .1).
(Here, (T,..,, p()) 1..., .1) is I-tuple of identity except the one at (j))
end if O (2200)n)) t 0 (2,000g)) + 0g (.,
end for
-4 1/t No 0
Ma O₂ (2,7T.))
Return
Algorithm Estimation of 0(y. (MM (theta_grad_est) 8e,
Input: E [[D]],
Measurement M, Parameters (1) E Xsigma, o (1) E Xpheta true label yt. je and a positive integers (()) N(0)
( O 22 0.0 for to
N) do
0a + universal (t.M, -, 01 (1, .1) (Algorithm 2
0111 (1) + end for for a = to N: do Bernoulli then
if (2.6g)
universal_estimateee (,, - 21,,,,,,,......
(Here, ट, E RR4 is the vector that has 1 in the jth component and O elsewhere.)
else
Oa (2,8,) + (-1) universal (p..M, J, d (1. .1)
end if O (2.0g)
- ((200) + 0
(2,8;)
end for
1 001)
(2,8,)
Return
-2
No
NGT
31

---

We proceed to demonstrate that both algorithms produce unbiased estimators, as stated in the following proposition.
Proposition A.17. gmg grad, est (Algorithm 3 and abeta_grast est (Algorithm) + provide unbiased estimators of Biv (all--r(MMe M.. and 01u (() respectively.
00g
Proof of Proposition A.17 We will show that Algorithm is an unbiased estimator of Then, by applying similar reasoning, demonstrate that the estimation produced by Algorithm
8ly ur M(MMd(())))
das
4 is also unbiased.
To start, let show E -4 y/
o o.!)
- 4 (v=- tr (MMMn())))) Let 0g and oa (2.
be the random variables defined Algorithm
Applying linearity of expectation, we obtain:
o )]
N
E
4 y/
of
-4 yr
Σ
N(
(
(1) A.16 we know E Oa
Applying Lemma
tr (Miln(pe)) Hence,
0.00) )
4 ( (MMdn(())))
No(h.1)
Continuing, we show that,
O.2.05m
E
Nt(iaa
(M(MMeis (Ai² (Db-11)))
(Milsss (Ai (p(2) pe, (P())))
Applying again linearity of expectation and incorporating the definition of oa (2.45.)) we arrive at:
E
O
(2))
N(
(2)
N ao
No(a.o)
univer universal M, J, 7. (T,....,)))
E
No
a=1
E
universal
universal_estimatto estimator (p.,M, b, (I, I, I, II.,,,....... .))]
2
32

---

By Lemma A.16 we know that,
(
tr (M (Ar (of--1)) - -" (Metiss (Ar (p() (es((())t))
((-)))
a=1
tr (Miless (Ai' (pt--1)))
No
N(a.9)
Σ tr (Miliss (Ai (P()) pi (ps((()))))
2
aml
2tt (Miliiss (Ai" (
(pt
f (Miless (Ai' (pL") pe (p)(t))))))
-4(y - 0 and o (3) are independent, combining results above, we obtain:
Since
O o.2
E
(10)) ]
((2001n)))
=E
V/
Nr(3-n)
(o(in))
17-00 ( (g 2 ppo Adppp) (pis)) )
(y) tr (MLLR(P))) -r (Mlli,s
Ad
A.3.3 Gradient Estimation via Probabilistic Subsampling: Algorithm and Approximation Lemma
By the definition of Algorithm I each step involves estimating the gradient of L. at (ō, 7) for given E [[D]]. To this end, we introduce the following estimation algorithm, which leverages stochastic sampling to estimate gradient w.r.t. both the inverse noise parameters á and PQC parameters 7.
33

---

Algorithm 5 Gradient Estimator betraddent_esttmeee
Input: Positive Integer t, Measurement M, Parameters 2 E Nsigmas 7 Xhheta true label
even positive integer N, and a binary vector (1., (llh), ,:: p!!) ... of length (d,, 0
Assumption: Assumption 3.2
Assumption 3.4
for / [L] do for K, do
if (b,))
then
L1 0. 8 8)) l,k
t si geagg_a____ssttt_.__. 7. 4.
else
Vhe प. 7 )))
) +-
1,k
end if
end for
end for
for I E
do
if (1)
then
VC. 7, 8 )) + gesta__rrd__sttt,AAA J, y. N/2, (21
else
vc &, 8 ))
end if end for Return
...., (V.. (L.. ( 7)
(7c, ( 7 .
)));
In the following lemma, we demonstrate that Algorithm produces an expected ht/p))aapproximatiie of the true gradient of L, (d, d).
Lemma A.18. Let E ZzI. gradient_ estimator (Algoritin 5 is (ap))aaaproximation of the true gradient of (.(d, 0) in expectation.
gradient__sttmatoon
Proof of Lemma A.78 For simplicity, define -(o, d) and |w]| to be the length of w. Let By the definition of Algorithm 5
B' (Bernoulli (1/p)) [oo] be a random binary vector in {0, 13 [w]
we note that its output can be represented by a random vector 7 @ VL, (w), where O is the element- wise vector multiplication and V, (w) is the estimation of the true full gradient V C, (w) generated by Algorithm 3 and Algorithm
Taking the expected value conditioned on p. we get:
E B (w) F (.... ((
Here, (VL, (w)) the th component of the vector VL, (w). From Proposition A.17 that E. (w) is unbiased, meaning that,
we know
E 7 + VL. (... (VL, (w)) VL. (w).
34

---

Taking the full expectation,
E p o
(w)
=
E
E T &.. v F.
Pr Pr
(0,
Σ
(FVVL( - Pr Pr
[E(0,1)lml
Σ pr[p] 7 @ Vv.(w))
FE(U,1)Im
((T))), ,(.,,BB())ii
3 VL, (w)
(,.,, =
11 .......
VL, (w)
-(1/p-..,1//)) 0 VV,
= (w)
P
Here, (0)a is the th component of the input vector.
Remark A. The scalar p Zz1 must exceed the total number of parameters (inverse noise parameters and PQC parameters) being optimized in order to guarantee that the expected density not O. particular, = 1, 7. is,
of the estimation given by E || o VL, 7. by definition, all-ones vector. Consequently, Paddiet_esstmaaee (Algorithm 5) provides unbiased estimate of the full gradient of L, at (d, 0).
A.4 Sample Complexity
review of Algorithm 5 iindicates that it employs stochastic selection strategy reduce the overall sample complexity. In the remainder of this section, we show that the total sample complexity decreases by factor of 1/p expectation. We begin by demonstrating that a mapping of the form 2n - X;) satisfies the bounded difference property.
C (y- X:) (
Proposition A.20. Let Z>o be some positive number. Let %i,... ..en {-1,1) be 2n independent random variables. Define {-1,1)2m - R by F(X1,... X2n) 2n
(y Ei=, X1) ( Eim++ Xs) for some constant C IR and y {-1,1). Then, for all
1 و j < 2n.
4C
max
,,,,,),,,,))))) F( .Y,,... X2n)
X; X,,X,E(-1.11
n
Proof of Proposition A.20 By the definition of F, we consider the following two cases:
35

---

Case 1: Suppose 1 S S n. Then, for any Xy, Xj E (-1, 1}, we obtain:
F(
-m.....
x2n)
F(X1: ... X,y'' -..... 22n)
1
Kjj
c
71
11
11
1 n
X
1
2n
C
y
X
Σ
X
n
12
imn+1
C
X
x
n1
1m++1
|C|
2n2
x,
Σ
[
(triangle inequality)
11
imn++
[C
.2 2
2/C|
n
n2
i=++1
Case 2: Suppose + S S 2n. Again, for any Xy E (-1,1}, we obtain:
F(X1,'.. Xn.... Ky.... X2n)
F(K,....
.Ky.... "Ky, x2n)
11
2n
X
X
X
71
71
72
i=n++ if1
1
X
-C
y
%i
X;
12
n1
X
X
[C]
X
1=1
2
S|C|
| + Σ
X
(triangle inequality)
11
1=1
2
4|C]
<IC| (1+
71
Together, we get
[||]]
4||
4|C n
max
"(X,,".. K,,... K2n) F(X,,.... ",,,'.. X2n) S max
n
A.4.1 Probabilistic Bound on Estimation Error for Gradient Estimators
In the following lemma, we establish a probabilistic bound for estimating partial derivative of C with deviation 6 >>
Lemma A.21. Let Jo Cu (7, d) and oe, Le (d, 8) denote the estimated partial derivative of E, with respect to وn4 and 02 at (ō, F). which are generated by Algorithum 3 or Algorithm 4 re- spectively. Similarly, let 0o Lg (2, 0 and 0o, Lr (δ, d) denote the corresponding true partial
36

---

derivatives. Then, for all O, the probabilitiess Pr do Le (3, 0o (2.
2
Bu, Li (7, 7) 2 $ are both upper-bounded by where I = exp { -2 EnEK, OLK and Ñ is a positive
and
Pr Jo Le
2(( ( 2
2 exp
32
even integer.
Proof of Lemma A.21 Li
We
begin
by
establishing the bound for Since r > O, we have
Pr
0, 2i
8.
2 E Dog Co d. θ
@o Li
(7, 8
E @o
Li S 0
2E
(1 @ay. Li
J f))
E D Li (,,7]] (13))
Thus, the probability can be written as:
Pr
@ La
d 0 E 20 Lt Φ 8 88] 2 2c[r]
PPr
(1Z))²
0osio Li
(d, 0 EE Dog f. ( 7
(rZ))² c|1]
=Pr ( . La ( 8) (Tz E jos,. C, (7. [T] > (r E
Next, we aim to show that the random variable (r 00 Lu ( 0 satisfies the bounded difference property. From the definitions in Algorithm re- re-written random map from T-raraal to IK defined the following:
we observe that 0o,,, Cr (o, f) can be
@orse Lx (
(d.0))
(1)
(1)
(,
(2.0
0i
N/2²
N/2
1/2
-(-4) y1
1
Σ
1
(2.02.)
N/2
N/2
8=1
3=N12+1
From the definitions in Algorithm we observe that both 0g and (2.0..))
(1)
are in the set {-I }z 0g
This implies that the random variables n 0g and r= -0g (2.0g.0) lie within 1-1, Consequently,
(1)
(2 (Dog. Ce (7. ))) becomes random map from 1-1, 13"h N
to R. For convenience, let us
re-express (17))2, (Do (Dos, Le (o. 7 as mapping F (-,,, - RR defined as follows:
N/2
N
F (x((F)
((f)
v(1) = (-4)
r y1
Σ ((
(P)
R/2"
N/2
N/2
0=N/2-1
where applying Proposition A.20 we observe the following bound on the difference when changing single input:
w/
-(-== -
=
0g for 1 < B E N/2, and w = r
(1)
0g
(2.0.a) for Ñ/2 + S E Ñ. By
|F (M
(x(7) ... .,, (... x(())
max
FFye((1111)
16
32
N/2
N
This shows that changing a single input alters the value of (T (0 Lu (7, B)) by at most meaning that (2=))2, (Don. Le (σ, ))) satisfies the bounded differences property. As result,
N
37

---

by applying McDiarmid's 's inequality [6], we know that for any >> 0, we deduce,
Pr (17)) )² 0oy,0 Lt
(Tz))2.
R 0os Ltt (v. 2 (r cc(
N
<2exp
-2 ((TT)) .,))²/ ,)²/
32/N
-2 ((T)))) 22 e))²² Ñ
exp
32
By Proposition A.17
we note that
0o,,0 Le 3 )} Dos, Ci (Z,0).
E
Therefore, obtain the following probability bound:
-2 )( N
Pr 0o, ca (3. -0o, ( (.. B) 2 20/0] s 2xx
322
Using similar arguments as above, we identify (rz))² 00; L, (7, as a random mapping from
{-1,1} N to R. From Proposition A. 16 N
we observe that changing a single input alters the value
of (I )² 0o, fe o. Ø)
by at most
32
Again, by applying McDiarmid's inequality and N
Proposition A.17 we obtain:
(1 - N
Pr Ou L. (Z, - -b,, (d, T) 2 s 2xep
322
Now returning the context of Algorithm 1 in each update round, exactly one d is sampled and as the vector of estimated partial derivatives generated by Algorithm 5 Given particular update round and p. the ah entry of O VC, 7, d) A 0 only when the a entry in D' is 1. Thus, to quantify the estimation error,
passed into Algorithm For simplicity, we define p VL, ब t
it suffices to measure the deviation between the non-zero entries of d V corresponding true partial derivative.
and their
Let
VL. d 7) denote the vector of the corresponding true partial derivatives. Here, d. 0 is the full true gradient of L. ( t). and the value of each entry in p
VL
VL. d. 0 is determined according the entries of p. In the following lemma, we estab-
p ② vc, (अ, p VC (7. (3.6)); with deviation
lish a probabilistic bound for conditioned on 7 and l.
Lemma A.22. Let p S
be a positive integer. For all > and Ñ (2Z)
Pr
p V
c t
P VoL. ( o 1) 2 c p
-2 22 .))) =)²N Ñ
< llb 2exp
322
where r r='= = =exp (-2 ΣKEK, oLK and 7 E (0, 13 (7,7))
38

---

For simplicity and clarity, we now denote (7. 8) by and the length of
Proof of Lemma A.22
A.22
by |w].
First, using the fact
|
S ||||| we note that | 26 =
| 2 e, which implies that:
Pr
+ vC (w) F VL, (w)
t, p
<
+ VL, (w) 7 o VL, (w)
(VL) - (w)) 2 2
= Prr
max
a=1," h/
p P)
Pr
3 E Ix (w)) - (VC, (w)) (VL,
2 t p
where 2 = fa E [|=:]]((] )a # 0}. Applying the union bound, we have:
Pr 3a E Izz (76 (wn) -VL. (w)) 2 2 E
Σ Pr ( (w)), (VL, (w)), 2 c|c aEL9
Then, since Le (w)) ) are generated by either Algorithm and Algorithm 4
by applying the
Lemma
A.2I we derive:
((T
- N
Pr (w)) (VL, (w)). 2 e/1 < 2exp
322
Combining all of the results above, obtain:
2 ((T))) )2.)) N
Pr p (w) F o vLL( 2 20/2,
Σ 2xxp
32
aET /
-2 22((T22) 2). -)²
-2 exp
322
By definition of Izz. note that [Iz| = 1 Therefore,
-2 ((T .E)
Pr (w (w))
2 2
2 P exp
322
(31)
A.4.2 Expected Bound on the Total Number of Measurements
In the following theorem, we present an expected bound for the total number of measurements
required to ensure that the deviation 7 o VL, σ. 8¹ D o VL, (7, 8 8)1l remains within
with probability at least - 8 over the distribution of p.
Theorem A.23. Given positive integer p < (d,7).. 0, and 8 E (0, 1), the expected total
number of measurements required for Algorithm 5 estimate the gradient of Le with deviation at
most E and probability 1 - over the distribution d' is at least
(7.8)
322
2
2 In
P
2 ( )2
8p
39

---

From Equation (3) Equation (8). and Equation TO).. we observe that the partial derivatives are asymptotically equal to (( for some æ E [-1, 1] when ( 7) is not too close to a stationary Proof of Theorem A.23 For simplicity, we again denote (अ. by w. where |w| represents its length. Taking the expected value of Equation 31) over the distribution of p., we observe:
Pr || O VL (w)
e vL. (w)
E Pr
+ VL (w) Đ
VL, (w)
2 2
Pr Pr
-2 22.)) N
pppl 1 1-1 1 - |
[
Σ
1
2xex
32
7E(U.1)1M
exp
2 2T z E)
[00
k (Iu))
(1
11
bu|-k
!
R
322
K=0
Next, Using the identity Ei= (1) 11p)) n/p for all n E Zz1 and p 2 1, we arrive at:
Pr P (w)
Vhe (1)) ||, 2 26/1]
s exp (((
wwl
(32)
322
Then, let us determine the expected number of measurements required to estimate each selected partial derivative (i.e., N) over the distribution of p'. Setting right hand side of Equation 32) to at most ó and solving for N, obtain:
32
2/wl 8p
In
2 2(T 2)
Moreover, since each entry of 7' is sampled from Bernoull((1/p) distribution, we know that
[|| l = [al In other words, there are /wl/pp number selected partial derivatives in expectation.
Combining the above results, we conclude that over the distribution of p', the expected total number of measurements (i.c., ([wl/p) NN required is at least:
[w]
32
2 2|w]
In
(33)
2 ( 32 e)²
8p
Remark A.24. Equation (33) suggests that small results in an exponential sample size. To mitigate this, one approach is to select è large enough to determine the direction of the gradient reliably.
To determine appropriate value for e, consider the value of the partial derivatives
point. Moreover, their directions are determined by
Thus, selecting E as r2g- for some x E (0, 1) suffices ensure a reliable determination of the gradient direction. Consequently, the expected total number of measurements required becomes at least:
w
322
w
In
(34)
2x²
8p
A.5 Convergence
A.5.1 Preliminary
We begin with review of fundamental results in calculus.
40

---

Proposition A.25 (Extreme Value Theorem [8 ). F X - Y be continuous, where Y is an ordered set the order topology. If X is compact, then there exist points e and in X such that F(c) S F(x) < F(d) for every x E X.
Proposition A.26 (Estimation of Lipschitz constant on C functions 13). Let C R" be an open subset, and suppose F U - IRm is of class C². Then F is Lipschitz continuous on every compact convex subset K U. The Lipschitz constant can be taken to be supzek (F)l(2 where (x)) is the Jacobian matrix.
Lemma A.27 (Decent Lemma). F R R is of class of C' and K is any compact and convex subset of Rd, then:
F(V) S F(x) VF(x)(v- - LK 11 xx12-
2
for all x,y E K, where Tgg is the Lipschitz constant.
Sketch Proof of Lemma A.27 By applying Proposition A.26 we know that is Lipschitz continuous on K. Furthermore, the Lipschitz constant LK is bounded by supzER R ||FF(x)]] Then, using the inequality established the proof Lemma 2.25 in 77 for any x, y E K, we know that:
Fly) & F(z) + (VF(x))) (v-x) + [
VV x)) - VF(x)]2 lly - x|12 dt.
Since K is convex, we know that x + Mly - x) K for all E [0, 1]. By the definition of local Lipschitz continuous, we can write:
VF (x + t(y z)) - VF(2)ll2 - Lx ||= + t(y - x) - =112 Lxt lly - x1|2 for all t E [0, 1]. Together, conclude that:
F(v) sF(x) + (VF(x))) (y-x) x) + S/¹ Txxl f1y - x01/ di
=F(x) + (VF(x))) (y (y-x) + Lx lly - -x xx1/ (Lo tdt
LK
=F(+) (VF(z)) (v- 2
lly x
Based on the results above, we present the following two properties of our objective function:
Lemma A A.28 and Lemma A.29
A.5.2 Properties of the Objective Function and Estimated Gradient
this section, we present several important properties of the objective functions & and C+G. defined in Section 3 as well as the properties of the estimated gradient obtained from Algorithm 5
Lemma A.28. Let L(d), Ø) be the function defined in Equation 6 On the compact set X Ksigma x Keheta L is Lipschitz continuous.
Proof of Lemma A.28 The proof of this lemma directly follows from Proposition A.26 Since maps ac
ac 0asa given in Equation (10) and 80, given in Equation (8) are defined on R 7
we know is
of class C². Then, applying Proposition A.26 know Z is Lipschitz continuous on X.
Lemma A.29. L(d) 0) 5( 7) is lower-bounded on R (.,8)
Proof of of A.29 Given the definition of [[ 7) and G( Z). we know that L( 7) G(d, ) is finite and well-defined when (7. 8) E X. So, it suffices to check L(d) 7 0) 8) lower bounded on X. Given the definition of and G, we know that L(7, 8) + 9( 8)
L( 8) X. Since L(d) 0) is continuous and X' is compact, by the extreme value theorem (Proposition A.25 we know L( L(d), 7) ) is lower-bounded, which implies L(7, 7) + प is lower-bounded.
41

---

Proposition A.30. Let D. & &c.. (7.0) be an estimation of gradient of Ci. at
some fixed point (Z, d) generated by Algorithm 5 where it uniform "1,.,,1P11
and Pt (Bernouli(L/pp)((
Then, v &c. - (3.Ø)
p. Fi. (σ.
1. VL d. 0 d) is bounded.
Proof of Proposition A.30 For clarity, let w denote (7. (Z.7). First, by taking the full expectation of
p. Ct (d. 0
using Lemma A.18 we obtain:
E p 2 VLa (w)
E
P O
VLn
(w)
VLi, (w)
1 VC (w),
D
[D] 1,=1
p
(35)
which leads to:
p 2 VLg (w) E
F. L
Vc, (wo)
p (1), - (2v (wo)
[0|]
( (10) )
E
[ 3 Ec (w) [
var
Here, (+), represents the th component of the input vector. From the observation of Algorithm 3
Algorithm and Algorithm note that p Vi. (wi) :f [-[ R defined by
V. ④ Wo. (w))
(o1,...
0N/2
NN/2+11"
O0
N/2
1
= -Draa -C
Yi. N/2
08
og
N/2 B=
2
where C is 2 and T exp {E,, j. Tzikk } and N is even number. Since the mapping
Veo E,, (w)) has finite domain, note that:
( (7.0 Vc. (w)) (01.-..,088 ,00,,.,, ,0,)
max og o(
( o Vir (10) (o1, ,o,, ... ,x)))))))))))))))))))
finite for all and Pe. Define
(B. VLn, (w)) (01, "",,".",,,,"." ,0,,..,,000 ,x)
max = 7l |=
(36)
- -(T 2 V (o.,:..,,,........ ... 00)))
42

---

Using the Efron-Stetin inequality, we obtain
p Vii (w)) )
E
p 3 J.., (w)) (01, ,0.,'., ,0z)
8=1
- -(7 2 VL, (w)) (0,, .. ,.. .. , )
N.12 2
1f
2
8=1
Together, we have
Ñ 122
V V. o VLc, (w)) s slu|-
2 < oc.
A.5.3 Convergence bound of the Learning Algorithm
We now present the convergence bound in the following theorem.
Theorem A.31. Let L( δ) + g 0) be the objective function, where C and G are defined in Equation (6) and Equation (7). Let { Wr = (2 T1)) ) ,t= L,... .t.,..,.....
be sequence of parameters generated by the proximal-SGD (Algorithm D. Suppose wn and w are the initial point and a stationary point in X Xasigma x Xihetas respectively. Let {n 0 < 1 < 1,1 0,... ,T} be a sequence such that Ei=o < 0c. Then,
p. 1w0 w/l2
3Lx Et= B Biaa mar
P PE V
Ei=1 Mh Σ nr [[(ea)]) [(w)) S
2 2Σ o
2 ' T)
2 2 1/
where V supw, V [ Tp.OVAL &L, (wr)). p is a positive integer, Lx is the local Lipschitz constant, and Braz= E (B(1.))2 2 + Σ |=1 (2((0) -20"" )²
Proof of Theorem A.37 For simplicity and clarity, we define wi ( B.). X Xaigma x
Xtheta, and D. ② Li, (w.) denotes the gradient estimation for Li, (wx) (Jir tr (MM)(t((ps)) generated by Algorithm Here, & uniform {1, 21.,..... [DI} and p. ~ (Bernoull (1/p)) /eee By definition, we know 1++1 proxg We 7/t P. @ VLG (wx) which implies that w; E X for all
t= ... T. Applying lemma 8.17 from [7], we obtain the following result:
|w1+1 - 3|2 proxg
Wi= Th De o Vi, (wz)) - proxg (w)
< 1i p. ② V.. (w.) - w 1
Using the fact ||0 - v112 = (x y)²² (x y) = ||ا11 2xT y + |1 we can expand the squared norm as below:
2
Wi 1/c pt o Cc. (wi) - w 2
w/ - T 21 The 2 Ei,, (we)) T
(w, w) + 1/ | Veo 0 VL, (.
Together, we have
||
w 2/2 |w: - 2| - (7. 3 Vi,, (w02) T
(w)-w)+il +i F
VL., (wx)
43

---

Now, taking the expectation conditioned on wte we get
||
- 11/ 1
2
2 w
T
2
<
-21
Dio 3 Eig (w=)) )
(we- (0--j 3)) +
D. C VL (we)
w
2
- 21p
pe. o A Ci (we)
(w: - w) w + n B
D. o Ec. (wre)
11
Wt
(37)
For the first term in Equation
(77.
using Proposition A.17 and linearity of gradient, we arrive at: (w, w)
(D.O Vig, (wx)
I
E
Wt
D
( + Vii, (wre)) T (we, w)) wre des b. 1) Pr 171)
E
E
[D]
7.
4=1
D
E
p 0 Li, (w:))² (u,, - w) -Pr
[
(0.
Σ
( G (V((w)))
(wi - w) Pr
Σ
De- Pr [Pal
o VL (w²)
(w:-w) w
1 (VL (w:))" (w, w.
p
For the second term in Equation 37. by Proposition A.30
we know that the value
p の Vca (wn:) conditioned on wy is bounded. This implies that E is also bounded, and specifically we have the inequality:
D. Vi, (wx) Wh
E
|| P.
G. (wi)
We SV
pe O VLa (wx)
Therefore, we have
[1lle] - w112 |/ure
w|12
21P
(VL (w:))² (w: w + 2 [e VL,, (wr))
We
S
p
By combining all the above and taking the full expectation, we get:
E ||---1 - क 2 l|w.
=E
EE
||++1
T122 - -1/w T1/ W1
277
SE
(VL(wt))) (we- - 3))
+ + V p: 3 GC, (wx)
p
27t (VL(w)))) (we- 3))
<E
+ v
p
where V = su V V. C Vi, (we)
Telescoping the both sides, we obtain:
T
E
w| |14-
w
[[
-E
WT
w/
|/w0 -m1
T
T
21k
(VL(u=)))) T (we - ww]
+ > m v
1=1
t=1
T
T
2ntt
E
(VL(w))) (w (w-
+ + n
(38)
p
tm1
44

---

Next, let us bound (V (VL(w)))) (w- - (w)) Using the fact that x"y ||a||| Ilvil cos & for any 2, we observe that:
(VL(w)) - VL(w))" - ww) - VL(w:) vl(w))l2 |/ well2-
On the left-hand side, we have:
(VL(w))) Vf ( - we) (VL(w))) (w- - ux) (VL(T)) ( we).
On the right hand side, since L is locally Lipschitz continuous on X' by definition, we know that:
[V)) VL(w)ll2 ||w well2 S Ixx ||w we
Combining and re-arranging terms, we get:
(VL(w:))) (w w;) (VL(w))² (w wx) + Lx l/ soee-welll.. well2
Furthermore, applying the decent lemma (Lemma A.27p. we have:
3Lx
(VL(w))) T w)) - L 17 [(wr) +
1 - well2 2
(39)
Now, combining Equation 38 and Equation 39 we obtain:
27
3Lx
WT Wwo w
((w) [(wr) +
1 we
+
n3 V.
2
which is equivalent to
Σ 2n
(E [[([w)]] [(w))
p
3Lx
$ W
w
E
|
well
n
p
3Lx
T
|| wo
1
+
E
well
V.
(40)
Since wi and w7 are both in X' and X' is hyper-rectangle, know that:
W well2 -
B((A))
Σ
B
28
B maz.
Im REA
Thus, Equation 400) can be written as:
Σ 2n (E [[[[(w))] - L(w)) I|wo w122t +
3Lx
Σ
B mar Σ
V
p
p
t=
2Σ
Finally, dividing both sides of the inequality by
obtain:
P
Σ mE [[wss)]] L(w) p p-llwp
w
3Lx ΣT" 1 B Bmar
p P niA
11=1
t=1
2
2 2Σ
Th
A.6
Additional Experiments
A.6.1
Evaluation on Fashion-MNIST Dataset
We evaluate our approach on the Fashion-MNIST dataset 32 samples; standard test split; three seeds). Fach x 28 image is standardized, reduced by PCA to compact feature vector (64 dimensions for our 6-qubit scenario), and fo-ormallzzed to unit length.
(Pullover vs. Shirt; 5000 training
45

---

In simulation, we directly initialize the statevector to the corresponding amplitude encoded state [x(x)), then apply a 6-qubit, two-layer HEA with parameterized Rx. Ry, Ry rotations and circular CNOT entangler as in Figure The labels are assigned to -1 (shirt) and +1 (pullover). This direct initialization avoids extra state preparation depth and associated gate noise, so our comparisons isolate training-time noise and mitigation effects. Besides, the initial learning rate (17(2) is selected using grid search across wide range and then refined with binary search for better precision. Subsequently, the learning rate is annealed monotonically.
Under static noise, our inverse learning method initially lags Van den Berg et al.'s pre-characterized approach but surpasses it mid-training, achieving lower final loss (Figure Sa). The noiseless PQC provides the best-case reference, while the baseline noisy PQC yields the highest error, confirming that both mitigation strategies effectively reduce noise impact. Consistent with these trends, Table summarizes the test accuracies of the compared approaches. Under dynamic noise, we emulate hardware drift by slightly increasing the eπrоr rates of three CNOT gates early in training. While Van den Berg et al.'s method retains its fixed calibration, our approach updates the inverse parameters online and achieves lower MSE by tracking the drift (Figure 5b). This highlights the practical advantage of adaptive inverse learning over pre-calibrated approaches in realistic quantum hardware conditions. Finally, probabiliistic subsampling analysis using the same static noise profile as Figure 5a reveals the expected convergence degradation with increased sparsity (p 1 > 2 4), with denser sampling providing more stable optimization, consistent with the MNIST results (Figure 5ch.
Table 2: Classification accuracies across Different methods for Fashion-MNIST dataset
ACCURACY
NOISELESS PQC
78.87
NOISY PQC (BASELINE)
69.57 4 3.35
MITIGATED PQC (VAN DEN BERG ET.AL)
72.37 2.53
OURS
74.23 2.84
A.6.2 Additional Results- PPC and ZNE under Dynamic Noise
Noise miligated PGC (Gurs)
2.00
Nnise mitigated POC (Ouns)
1.75
PEC miligation
PEC milipation
1.75
miligaioe
1.50
Toirinn bas (45E)
Training bas (MSE)
1.25
1.00
1.00
0.75
0.75
0.50
0.25
1.5
20
25
30
30
40
50
Epochs
Epochs
Using MNIST dataset (3
Using Fashion-MNIST dataset (Pullover Shirt)
Figure 6: Training loss comparison among PEC, ZNE and our mitigation framework for binary classification task under dynamic noise. Here, each epoch summarizes 50 iterations of the training process.
Under dynamic noise, our adaptive inverse noise learning method updates the inverse noise parameters every iteration and consistently achieves the lowest final MSE on both MNIST (Figure (6a) and Fashion- MiShIT (Figure (b). PEC often looks competitive because it applies the full inverse from pre-calibrated noise map, but has practical limits: (i) measurement intensive: the shot demand multiplies across mitigated gates, leading to an effective exponential growth with circuit depth; (ii) does not track drift without recalibration; and (iii) it loses accuracy as the device moves further away from the calibration point. ZNE performs worst on average with higher variance, because
46

---

extrapolation assumes stable noise scaling. which breaks under time-varying, noise. The convergence of our method across both datasets supports the conclusion that adaptive inverse noise learning is important for stable optimization on realistic, drifting hardware- capabilities that neither fixed PEC calibrations nor Z.N''s static extrapolation provide without manual intervention.
47


Reviews:
[{"review_id": "F4smHek7dp", "reviewer": "Reviewer_BRsB", "summary": "This 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.", "questions": "- 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?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\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", "was_revised": true, "final_justification": "This paper presents a scalable joint optimization approach to mitigate Pauli noise in PQC training, with a novel gradient estimation method, supporting theory, and empirical results showing advantages over baselines. I would like to recommend acceptance for this work.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and insightful comments. \n\nThe 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.\n\nWith 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.\n\n## Responses to specific questions:\n\n\n### Q1: Of the assumptions you list, which do you think will pose the biggest challenge for applying your method to real hardware?\n\nThe 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.\n\n\n### 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?\n\nWe 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.\n\n\n\n### References:\n[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.\n\n[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.\n\n[3] Gonzales, A., Shaydulin, R., Saleem, Z. H., & Suchara, M. (2023). Quantum error mitigation by Pauli check sandwiching. Scientific Reports, 13(1), 2122."}, {"review_id": "SpBIFWwFaB", "reviewer": "Reviewer_qjyC", "summary": "Parameterized 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.", "questions": "1. 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?", "limitations": "I 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "Having reviewed all the final points, i believe this approach is still conceptually novel and perhaps useful in the near term.  I believe adding the conditions on the inverse operator to the main text will be useful in improving clarity, but perhaps uses of the method in fault tolerance still remain unclear.   Overall I would lean towards accepting this paper.", "rebuttal": "Many thanks for the thoughtful and insightful comments. \n\nWith 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.\nNotably, 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.\n\nThere 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.\n\n## Responses to specific questions:\n\n### 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.\n\nIn 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: \n$$\n\\Lambda^{-1}(\\rho) = q \\rho - (1 - q) P \\rho P,\n$$\nand then \n$$\n\\mathrm{tr}(\\Lambda^{-1}(\\rho)) = q \\, \\mathrm{tr}(\\rho) - (1 - q) \\, \\mathrm{tr}(P \\rho P) = 2q - 1,\n$$ \nwhich 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 $\\rho$.\n\nHowever, 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:\n\n$$\n\\Lambda_l^{-1} \\circ \\Lambda_l = \\bigcirc\\_{k \\in \\mathcal{K}\\_l} \\left( \\frac{1 + e^{2(\\sigma\\_{l,k} - \\lambda\\_{l,k})}}{2} I \\cdot I + \\frac{1 - e^{2(\\sigma\\_{l,k} - \\lambda\\_{l,k})}}{2} P\\_k \\cdot P\\_k \\right),\n$$\nwhere $\\lambda_{l,k}$ 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.\n\nWe are happy to add this discussion to the manuscript.\n\n\n### 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?\n\nThe 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.\n\nWe 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:\n$$\n\\mathcal{E} \\circ \\Lambda = (1 - q - w + 2 w q) I \\cdot I + (q + w - 2 w q) P \\cdot P.\n$$\nFor this composition to equal the identity channel, we must have:\n$$\nq + w - 2 w q = 0.\n$$\nThis 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.\n\n\n### 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?\n\nWhile 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."}, {"review_id": "yImSaevsnw", "reviewer": "Reviewer_sqmh", "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.", "questions": "1.\tPlease see the Weaknesses.\n2.\tCan the proposed method be applied to multi-class classification tasks?\n3.\tHow 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", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n•\tJoint optimization of PQC and noise mitigation parameters is novel and well-justified.\n•\tTheoretical derivations and results are illustrated in a rigorous manner.\n•\tExperimental results demonstrate the proposed method’s strong performance.\nWeaknesses:\n•\tStrong assumptions are applied, assuming idealized conditions such as noiseless Pauli operations and the absence of SPAM errors.\n•\tExperiments are limited to the MNIST dataset, potentially limiting the generalibility of the proposed method.\n•\tMissing ablations on the locality parameter c.", "was_revised": true, "final_justification": "I have carefully considered author responses and also other reviews. I thank the authors for addressing some of my concerns. This is a good paper. I think I will keep my score.", "rebuttal": "Many thanks for the thoughtful and insightful comments.\n\n## Responses to specific questions:\n\n### Q1: Addressing Weaknesses\n\n### Strong assumptions are applied, assuming idealized conditions such as noiseless Pauli operations and the absence of SPAM errors.\n\nRegarding 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.\n\nOther 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.\n\nWith 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.\n\n### Experiments are limited to the MNIST dataset, potentially limiting the generalibility of the proposed method.\n\nWe acknowledge the limitation of focusing solely on MNIST, and we plan to evaluate our approach on more complex datasets in the final manuscript.\n\n### Missing ablations on the locality parameter c.\nWe 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:\n\n  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.\n\n   ii) Two-local gates tend to experience lower noise, which makes the optimization problem more tractable.\n\n   iii) Two-local gates are more practical to implement on current NISQ devices.\n\n\n### Q2: Can the proposed method be applied to multi-class classification tasks?\n\nOne 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.\n\n### 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?\n\nAs 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.\n\nWe thank the reviewer again for the constructive feedback and the opportunity to clarify these points.\n\n### References:\n[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.\n\n[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.\n\n[3] Gonzales, A., Shaydulin, R., Saleem, Z. H., & Suchara, M. (2023). Quantum error mitigation by Pauli check sandwiching. Scientific Reports, 13(1), 2122."}, {"review_id": "NDhYSMttdK", "reviewer": "Reviewer_cSqD", "summary": "This 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.", "questions": "I hope the authors to address the major concerns raised in the Strengths And Weaknesses part.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "From 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.", "was_revised": true, "final_justification": "I thank the authors for the follow-up explanations. I think the authors addressed my concerns by pointing out that the absence of barren plateaus is not necessary and showing how to address the SPAM errors. Given that the authors will add the numerical comparison and remove the unnecessary assumptions in the revised manuscript, I believe this work a natural fit for NeurIPS 2025 and will raise my score accordingly.", "rebuttal": "We thank the reviewer for the thoughtful critique.\n\n## Responses to specific questions:\n\n### 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.\n\nWhile the strategies proposed in prior efforts have demonstrated potential, they suffer from the following key limitations.\n\nFirst, 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.\n\nSecond, 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.\n\nOur 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.\n\n### 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.\n\nRegarding 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.\n\nWith respect to barren plateaus, all gradient based optimizers suffer from this problem. Specifically, barren plateaus are characterized by the conditions\n$$\n    \\mathbb{E}[\\partial\\_\\theta C] = 0 \\quad \\text{and} \\quad \\text{Var}[\\partial\\_\\theta C] \\leq \\mathcal{O}(1/b^n)\n$$\nfor 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.\n\nFinally, 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.\n\n### References: \n[1] Cincio, L., Rudinger, K., Sarovar, M., & Coles, P. J. (2021). Machine learning of noise-resilient quantum circuits. PRX Quantum, 2(1), 010324.\n\n[2] Sharma, K., Khatri, S., Cerezo, M., & Coles, P. J. (2020). Noise resilience of variational quantum compiling. New Journal of Physics, 22(4), 043006.\n\n[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.\n\n[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.\n\n[5] Bittel, L., & Kliesch, M. (2021). Training variational quantum algorithms is NP-hard. Physical review letters, 127(12), 120502."}]

Author rebuttal:

Meta-review: This paper proposed a gradient-based framework for learning parameterized quantum circuits (PQCs) assuming noise exists in quantum gates. In theory, this paper proved a sample complexity bound on estimation of gradients, as well as convergence guarantee of optimization the loss function by gradient descents. Numerical results demonstrate that the proposed method is better than previous quantum error mitigation methods such as Van Den Berg et al.'s Sparse Pauli Lindblad Method.

Reviewers found the studied problem and model of practical relevance. For the initial version of reviews, there are concerns about the assumptions on theory results (Assumptions 3.1-3.5) and settings of the experiments. During the rebuttal, the authors made detailed explanations and promised various changes, including that the c-locality and barren plateau assumptions can be removed because they do not affect the mathematical derivations themselves, and more experiments will be added for more complex datasets as well as for PEC/ZNE baselines. The reviewers then converge to accept this paper unanimously. 

In the final version of the paper, the authors should consider to add:
- Polish the theory parts to remove unnecessary assumptions especially those on barren plateau, and make a clearer discussion on which assumption are definitely necessary to prove the main theorems (Theorem 3.1 and Theorem 3.2);
- Add more experiments for more complex datasets as well as for PEC/ZNE baselines;
- This was not mentioned in reviews, but when the AC checked the paper, it is observed that Theorem 3.2 is a bound that didn't specify the learning rate. It would be helpful for the authors to explain 1) how in theory we shall choose the learning rates to optimize the convergence rate; and 2) how are learning rates chosen in the experiments in Section 4.
- A typo to fix: Line 154, a inverse -> an inverse
Decision: Accept (poster)