Mitigating Barren Plateaus in Quantum Neural Networks via an AI-Driven Submartingale-Based Framework

## Abstract

Abstract In the era of noisy intermediate-scale quantum (NISQ) computing, Quantum Neural Networks ( qnn s) have emerged as a promising approach for various applications, yet their training is often hindered by barren plateaus (BPs), where gradient variance vanishes exponentially in terms of the qubit size. Most existing initialization-based mitigation strategies rely heavily on pre-designed static parameter distributions, thereby lacking adaptability to diverse model sizes or data conditions. To address these limitations, we propose AdaInit, a foundational framework that leverages generative models with the submartingale property to iteratively synthesize initial parameters for qnn s that yield non-negligible gradient variance, thereby mitigating BPs. Unlike conventional one-shot initialization methods, AdaInit adaptively explores the parameter space by incorporating dataset characteristics and gradient feedback, with theoretical guarantees of convergence to finding a set of effective initial parameters for qnn s. We provide rigorous theoretical analyses of the submartingale-based process and empirically validate that AdaInit consistently outperforms existing initialization methods in maintaining higher gradient variance across various qnn scales. We believe this work may initiate a new avenue to mitigate BPs.

## 1 Introduction

Figure: Figure 1: Example of BPs’ mitigation process. A plateau-dominated loss landscape ($1^{st}$ image), a.k.a. BPs, could be gradually recovered to the normal case ($3^{rd}$ image) after mitigation.
Refer to caption: x1.png

In recent years, there have been significant advancements in quantum computing, particularly with the advent of noisy intermediate-scale quantum (NISQ) devices Preskill (2018). Within this research landscape, quantum neural networks (qnns), which integrate quantum circuits with classical deep-learning layers, have been widely applied in various domains, such as quantum machine learning Stein et al. (2021), quantum chemistry and materials modeling Kandala et al. (2017), and combinatorial optimization Farhi et al. (2014). However, recent studies Ortiz Marrero et al. (2021); Cerezo et al. (2021) reveal that the performance of qnns may be hindered due to gradient issues, such as barren plateaus (BPs), referring to a kind of gradient issue that the initialization of qnns might be trapped on a flattened landscape at the beginning of training. McClean et al. (2018) first systematically investigate BPs and affirm that the gradient variance will exponentially decrease when the qnns satisfy the assumption of the 2-design Haar distribution. Under these circumstances, most gradient-based training approaches would fail.
To better illustrate the BPs’ mitigation process, we present an example in Fig. [1](https://arxiv.org/html/2502.13166v2#S1.F1).

Numerous studies have been devoted to mitigating the BPs, among which initialization-based strategies have proven particularly effective by using well-designed distributions to initialize qnns’ parameters Sack et al. (2022). However, most existing initialization-based strategies, such as GaInit Zhang et al. (2022) or BeInit Kulshrestha & Safro (2022), rely on a one-shot generation of static initial parameters, which suffer from two critical limitations: (i) they often depend on idealized distribution assumptions, and (ii) they lack adaptability when scaling to various model sizes or data conditions. These limitations are particularly problematic where BPs hinder effective qnns’ training.

To address the limitations, we propose AdaInit, an iterative framework that combines generative modeling with the submartingale property. We prove its convergence, demonstrating that AdaInit Adaptively generates effective Initial parameters for qnns. These parameters ensure non-negligible gradient variance at the beginning of training, thereby mitigating BPs. Beyond the theoretical guarantee, AdaInit intuitively can help qnns locate initial parameters in the “non-flat” loss landscape, which prevents training from being trapped from the start.
Our method is grounded by two key ideas.
First, instead of statically pre-designing the initialization distribution, we leverage generative models, such as large language models (llms), to synthesize candidate parameters based on dataset descriptions and prior gradient feedback. This allows the initializer to actively explore the parameter space and adaptively refine the candidate.
Second, by modeling the iterative process as a submartingale, we provide a theoretical guarantee that this process will almost surely converge within a finite number of iterations to initial parameters that yield non-negligible gradient variances. We model such a process because the submartingale property can depict the trend of expected improvement in gradient variance and guarantee convergence in finite time. Besides theoretical analysis, we conduct extensive experiments to validate the effectiveness of our proposed framework across various model scales. The results reveal that our framework can maintain higher gradient variances against three classic initialization methods and two popular initialization-based strategies for mitigating BPs.
Overall, our primary contributions can be summarized as:

- •
We propose a new artificial intelligence (AI)-driven submartingale-based framework, AdaInit, for mitigating BPs. To the best of our knowledge, we open a new avenue to leverage llms with submartingale property to model qnns’ initial parameters for mitigating BPs.
- •
We theoretically analyze the submartingale property of the iterative process and rigorously prove its supremum and expected hitting time.
- •
Extensive experiments across various model scales demonstrate that as the model size of qnns increases, our framework can maintain higher gradient variances against classic initialization methods.

## 2 Related Work

McClean et al. (2018) first investigated barren plateau (BP) phenomena and demonstrated that under the assumption of the 2-design Haar distribution, gradient variance in qnns will exponentially decrease to zero at the beginning of training as the model size increases. In recent years, enormous studies have been devoted to mitigating BP issues in qnns Qi et al. (2023). Cunningham & Zhuang (2024) categorize most existing studies into the following five groups.
(i) Initialization-based strategies initialize model parameters with various well-designed distributions in the initialization stage Grant et al. (2019); Sack et al. (2022); Mele et al. (2022); Grimsley et al. (2023); Liu et al. (2023); Park & Killoran (2024); Kashif et al. (2024); Shang & Shi (2025).
(ii) Optimization-based strategies address BP issues and further enhance trainability during optimization Ostaszewski et al. (2021); Suzuki et al. (2021); Heyraud et al. (2023); Liu et al. (2024); Sannia et al. (2024). (iii) Model-based strategies attempt to mitigate BPs by proposing new model architectures Li et al. (2021); Bharti & Haug (2021); Du et al. (2022); Selvarajan et al. (2023); Tüysüz et al. (2023); Kashif & Al-Kuwari (2024). (iv) To address both BPs and saddle points, Zhuang et al. (2024) regularize qnns’ model parameters via Bayesian approaches.
(v) Rappaport et al. (2023) measure the BP phenomenon via various informative metrics.

## 3 Preliminaries

In this section, we first introduce the preliminary background about the basics of quantum computing and barren plateaus, and then present the necessary tools from probability theory.

Quantum Basics.
A quantum state can be seen as a unit vector $|\psi\rangle$ in a complex Hilbert space $\mathcal{H}^{m}$, satisfying the normalization condition $\langle\psi|\psi\rangle=1$. We use the Dirac bra-ket notation, where ket $|\psi\rangle$, bra $\langle\psi|$ denote a column vector in $\mathbb{C}^{m}$ and its Hermitian conjugate (conjugate transpose), respectively.
Any $|\psi\rangle$ can be written as a linear combination of computational basis states, $|\psi\rangle=\sum_{i=1}^{m}c_{i}|i\rangle$, where $c_{i}\in\mathbb{C}$ are called the *amplitudes* of the basis states $|i\rangle$.
Given two states $|\psi\rangle\in\mathcal{H}^{m}$ and $|\phi\rangle\in\mathcal{H}^{n}$, their inner product can be denoted by $\langle\psi|\phi\rangle\triangleq\sum_{i}\psi^{\dagger}_{i}\phi_{i}$, whereas their tensor product can be denoted by $|\psi\rangle\otimes|\phi\rangle\in\mathcal{H}^{m\times n}$.
If we measure state $|\psi\rangle=\sum_{i=1}^{m}c_{i}|i\rangle$ on a computational basis, we will obtain $i$ with probability $|c_{i}|^{2}$ and the state will collapse into $|i\rangle$ after measurement for all $i$. A quantum algorithm works by applying a sequence of unitary operators to an initial state.

Variational Quantum Circuits (vqcs) play a core role in quantum neural networks (qnns). Typical vqcs consist of a finite sequence of unitary gates $U(\bm{\theta})$ parameterized by $\bm{\theta}\in\mathbb{R}^{LNR}$, where $L$, $N$, and $R$ denote the number of layers, qubits, and rotation gates. $U(\bm{\theta})$ can be formulated as:

$$ $U(\bm{\theta})=U(\theta_{1},...,\theta_{L})=\prod_{l=1}^{L}U_{l}(\theta_{l}),$ (1) $$

where $U_{l}(\theta_{l})=e^{-i\theta_{l}V_{l}}$, $V_{l}$ is a Hermitian operator.

qnns, which are built by wrapping neural network layers with vqcs, can be optimized using gradient-based methods. To optimize qnns, we first define the loss function $E(\bm{\theta})$ of $U(\bm{\theta})$ as the expectation over Hermitian operator $H$:

$$ $E(\bm{\theta})=\langle 0|U(\bm{\theta})^{\dagger}HU(\bm{\theta})|0\rangle.$ (2) $$

Given the loss function $E(\bm{\theta})$, we can further compute its gradient by the following formula:

$$ $\partial_{k}E\equiv\frac{\partial E(\bm{\theta})}{\partial\theta_{k}}=i\langle 0|U_{-}^{\dagger}\left[V_{k},U_{+}^{\dagger}HU_{+}\right]U_{-}|0\rangle,$ (3) $$

where we denote $U_{-}\equiv\prod_{l=0}^{k-1}U_{l}(\theta_{l})$ and $U_{+}\equiv\prod_{l=k}^{L}U_{l}(\theta_{l})$. Also, $U(\bm{\theta})$ is sufficiently random s.t. both $U_{-}$ and $U_{+}$ (or either one) are independent and match the Haar distribution up to the second moment.

Barren Plateaus (BPs) are first investigated by McClean et al. (2018), who demonstrate that the gradient variance $\mathrm{Var}[\partial E]$ of qnns at the beginning of training will exponentially decrease as the number of qubits $N$ increases when the random qnns match 2-design Haar distribution. This exponential pattern can be approximated as:

$$ $\mathrm{Var}[\partial E]\propto 2^{-2N}.$ (4) $$

The [Eq. 4](https://arxiv.org/html/2502.13166v2#S3.E4) indicates that $\mathrm{Var}[\partial E]$ will approximate zero when the number of qubits $N$ is very large, i.e., most gradient-based approaches will fail to train qnns in this case.

Based on the above description, we formally state the problem that we aim to solve as follows:

###### Problem 1 .

By leveraging a generative AI (GenAI) model, such as an llm, we refine posterior with adaptive prompting, i.e., iteratively generate effective initial parameters $\bm{\theta^{*}_{0}}$ for a qnn that yields non-negligible gradient variance $\mathrm{Var}[\partial E]$, thereby mitigating barren plateaus (BPs).

Tools from Probability Theory. Below, we review the definition of martingale (submartingale), along with key tools relevant to our work. We adapt the descriptions from Williams (1991); Freeman & Stephenson (2025).

###### Definition 1 (Martingale, Williams ( 1991 ) ) .

Let $\{M^{(t)}\}_{t\geq 1}$ be a stochastic process w.r.t. a filtration $\{\mathcal{F}^{(t)}\}_{t\geq 1}$ on a probability space $(\Omega,\mathcal{F},P)$. The process $\{M^{(t)}\}$ is called a *martingale* if (i) $\{M^{(t)}\}$ is adapted, (ii) $\mathbb{E}[|M^{(t)}|]<\infty$, for $\forall\ t\in\mathbb{Z}^{+}$, (iii) $\mathbb{E}[M^{(t+1)}\mid\mathcal{F}^{(t)}]=M^{(t)}$, almost surely for $\forall\ t\in\mathbb{Z}^{+}$.

If (iii) is replace by $\mathbb{E}[M^{(t+1)}\mid\mathcal{F}^{(t)}]\geq M^{(t)}$ almost surely, we say $\{M^{(t)}\}$ is a *submartingale*.

###### Theorem 1 (Doob’s Forward Convergence Theorem, Williams ( 1991 ) ) .

Let $\{M^{(t)}\}_{t\geq 1}$ be an $L^{1}$-bounded *submartingale* (in Def. [1](https://arxiv.org/html/2502.13166v2#Thmdefinition1)). Then, almost surely, the limit $M^{(\infty)}=\lim_{t\to\infty}M^{(t)}$ exists and is finite.

###### Theorem 2 (Doob’s Optional Stopping Theorem, Williams ( 1991 ) ) .

Let $\{M^{(t)}\}_{t\geq 1}$ be a *submartingale* (in Def. [1](https://arxiv.org/html/2502.13166v2#Thmdefinition1)) and let $\tau$ be a stopping time. Then $\mathbb{E}[M^{(\tau)}]\geq\mathbb{E}[M^{(0)}]$ if any one of the following conditions hold: (i) $\tau$ is bounded, (ii) $P[\tau<\infty]=1$ and $\{M^{(t)}\}$ is bounded for $\forall\ t\in\mathbb{Z}^{+}$, (iii) $\mathbb{E}[\tau]<\infty$ and $|M^{(t)}-M^{(t-1)}|$ is bounded for $\forall\ t\in\mathbb{Z}^{+}$.

###### Theorem 3 (Dominated Convergence Theorem, Williams ( 1991 ) ) .

Let $M^{(t)}$, $M$ be random variables s.t. $M^{(t)}\to M$ almost surely. There exists a random variable $Y\in L^{1}$ s.t. for $\forall\ t\in\mathbb{Z}^{+}$, $|M^{(t)}|<Y$ almost surely, then $\mathbb{E}[M^{(t)}]\to\mathbb{E}[M]$ as $t\to\infty$.

###### Lemma 1 (Minimum of Stopping Times, Freeman & Stephenson ( 2025 ) ) .

Let $\tau_{1}$ and $\tau_{2}$ be stopping times w.r.t. a filtration $\{\mathcal{F}^{(t)}\}$. Then $\tau_{1}\wedge\tau_{2}=\min(\tau_{1},\tau_{2})$ is also a $\{\mathcal{F}^{(t)}\}$ stopping time.

###### Problem 1 .

###### Definition 1 (Martingale, Williams ( 1991 ) ) .

###### Theorem 1 (Doob’s Forward Convergence Theorem, Williams ( 1991 ) ) .

###### Theorem 2 (Doob’s Optional Stopping Theorem, Williams ( 1991 ) ) .

###### Theorem 3 (Dominated Convergence Theorem, Williams ( 1991 ) ) .

###### Lemma 1 (Minimum of Stopping Times, Freeman & Stephenson ( 2025 ) ) .

## 4 Our Proposed Framework

Figure: Figure 2: Our proposed framework follows an iterative process over $T$ iterations (gray area). In $t$-th iteration, we perform four sequential steps: (i) Generate $\bm{\theta}_{0}^{(t)}$ using a Gen AI model, $f(\cdot)$, (ii) Compute $\mathrm{Var}[\partial E^{(t)}]$ after qnn’s training, (iii) Calculate EI, $\Delta^{(t)}$, and (iv) Update prompts $x_{p}^{(t+1)}$, historical maximum gradient variance $S^{(t)}$, and effective candidates $\bm{\theta}_{0}^{*}$ for next iteration. Dashed arrows indicate data flow and corresponding outputs in each step.
Refer to caption: x2.png

In this study, we introduce a new framework, AdaInit, designed to mitigate BP issues in qnns by leveraging generative AI (GenAI) models, particularly llms. Our key innovations can be described as follows.
(i) First, unlike conventional one-shot initialization strategies, we propose a generative approach that iteratively generates effective initial model parameters $\bm{\theta^{*}_{0}}\in\mathbb{R}^{LNR}$ for qnns that yield non-negligible gradient variance $\mathrm{Var}[\partial E]$, thereby mitigating BPs. In each iteration, we employ an llm to refine the posterior (candidate initial model parameters $\bm{\theta_{0}}$) through adaptive prompting. After the posterior refinement, we train the qnn initialized with the generated $\bm{\theta_{0}}$ and further compute its $\mathrm{Var}[\partial E]$ in the early stage of training for monitoring BPs. The benefit of using llm to refine the posterior is that the llm can incorporate diverse textual instructions via prompts and adaptively update the prompts based on feedback from the previous iteration. This adaptive refinement allows our framework to dynamically optimize the generation process.
(ii) To validate the generation quality, we employ Expected Improvement (EI), $\Delta^{(t)}$, as a guiding metric for the iterative process. Furthermore, we rigorously prove that the process satisfies the properties of a submartingale. Consequently, we theoretically establish the boundedness, thereby demonstrating that our proposed framework will ultimately find effective initial model parameters for qnnsin a finite step.

Figure: Algorithm 1 Iteratively generate effective initial parameters for qnns using a generative model.

We present our framework workflow in Fig. [2](https://arxiv.org/html/2502.13166v2#S4.F2) and further introduce details in Algo. [1](https://arxiv.org/html/2502.13166v2#alg1). Given a GenAI model $f(\cdot)$, prompts $x_{p}$ for the $f(\cdot)$, a qnn $g(\cdot)$, and the number of iterations $T$, we first initialize $f(\cdot)$, $x_{p}$ (line 1) and also create an empty list $\varnothing$ for $\bm{\Theta}_{0}^{*}$ to collect candidates of qnn’s initial model parameters (line 2). After initialization, we run $T$ iterations for generation (line 3). In each iteration, let’s say in the $t$-th iteration, we first employ $f(\cdot)$ with prompts $x_{p}^{(t)}$ and a prior distribution $P(\bm{\theta}_{0}^{(t)})$ to refine the posterior distribution $P(\bm{\theta}_{0}^{(t)}|x_{p}^{(t)})$, which is the generated initial model parameter $\bm{\theta}_{0}^{(t)}$ for the qnn (line 4). After generation, we train the qnn $g(\bm{\theta}_{0}^{(t)})$ with certain training epochs and compute the gradient variance $\mathrm{Var}[\partial E^{(t)}]$, whose gradient is abbreviated from $\frac{\partial E(\bm{\theta}^{(t)})}{\partial\bm{\theta}^{(t)}}$, where $\bm{\theta}^{(t)}$ denotes the qnn’s model parameter in the $t$-th iteration (line 5). After computing the variance, we evaluate the improvement using the Expected Improvement (EI) metric, comparing the current gradient variance $\mathrm{Var}[\partial E^{(t)}]$ to the historical maximum gradient variance, which is the cumulative sum of EI when EI meets the following conditions (line 6). If the current EI, $\Delta^{(t)}$, is effectively improved, i.e., $\Delta^{(t)}\geq\nicefrac{{1}}{{(poly(N,L)K)}}$, where $\nicefrac{{1}}{{(poly(N,L)K)}}$ (with a parameter $K$ be determined later) denotes a strictly positive lower bound on the gradient variance of an $N$-qubit, $L$-layer qnn for each iteration, in the absence of BPs (line 7), then we update the prompts for the next iteration based on the current initial model parameters $\bm{\theta}_{0}^{(t)}$, the current gradient variance $\mathrm{Var}[\partial E^{(t)}]$, and the historical maximum gradient variance $S^{(t-1)}$ (line 8). After updating prompts, we update the historical maximum $S^{(t)}$ for the next iteration, where (if $\Delta^{(t)}\geq\nicefrac{{1}}{{(poly(N,L)K)}}$) $S^{(t)}=S^{(t-1)}+\Delta^{(t)}=\mathrm{Var}[\partial E^{(t)}]$ (line 9) and further concatenate $\bm{\theta}_{0}^{(t)}$ to the candidate list $\bm{\Theta}_{0}^{*}$ (line 10), which will be returned at the end (line 13). If so, the most effective initial model parameter $\bm{\theta}_{0}^{*}$ will be the last element in the candidate list. We can see that the input parameter $K$ is somewhat related to a wanted increment in $\mathrm{Var}[\partial E^{(t)}]$, which is further linked to the desired underlying property provided by the GenAI model. This connection is explicitly shown in the theoretical analysis below.

### Analysis of time and space complexity.

Our framework runs $T$ iterations. In each iteration, posterior refinement, which is linearly related to the output size of $\bm{\theta}_{0}$, takes $\mathcal{O}(|\bm{\theta}_{0}|)$ for a fixed-size qnn. Besides, training $g(\bm{\theta}_{0})$ with $T_{tr}$ epochs may take $\mathcal{O}(T_{tr}\cdot|\bm{\theta}_{0}|)$, where $T_{tr}$ denotes the number of training epochs for qnn. Combining $\bm{\theta}_{0}\in\mathbb{R}^{LNR}$, the total time complexity is $\mathcal{O}(T\cdot(L\cdot N\cdot R+T_{tr}\cdot L\cdot N\cdot R))\approx\mathcal{O}(T\cdot T_{tr}\cdot L\cdot N\cdot R)$.
The space complexity primarily depends on the storage requirements. $\bm{\Theta}_{0}^{*}$ at most stores $T$ number of $\bm{\theta}_{0}$, which consumes $\mathcal{O}(T\cdot|\bm{\theta}_{0}|)$. The output of posterior refinement takes $\mathcal{O}(|\bm{\theta}_{0}|)$ space. Gradient variance and EI are scalars, which cost $\mathcal{O}(1)$ space. The prompts $x_{p}$ are iteratively updated and thus occupy $\mathcal{O}(|x_{p}|)$ space. Considering the size of $\bm{\theta}_{0}$, the total space complexity is $\mathcal{O}(T\cdot L\cdot N\cdot R+L\cdot N\cdot R+|x_{p}|)\approx\mathcal{O}(T\cdot L\cdot N\cdot R+|x_{p}|)$.

### Theoretical analysis of our framework.

We first present some necessary results and further discuss how those results can be interpreted. Full Proofs can be found in the Appendix [A](https://arxiv.org/html/2502.13166v2#A1).

First, we define the Expected Improvement (EI) at each iteration $t$ as $\Delta^{(t)}$ and its cumulative sum in the past iterations as $S^{(t-1)}$ in Def. [2](https://arxiv.org/html/2502.13166v2#Thmdefinition2). Besides, we assume that the maximum possible gradient $\partial E_{max}$ during qnn’s training is bounded by a positive constant $B_{\partial E}$, which is practical in real-world simulation. Next, we establish an upper bound for EI through Lem. [2](https://arxiv.org/html/2502.13166v2#Thmlemma2) and Lem. [3](https://arxiv.org/html/2502.13166v2#S4.Ex4).

###### Definition 2 (Expected Improvement) .

For $\forall$ $t\in\mathbb{Z}^{+}$, the Expected Improvement (EI) in the $t$-th iteration is defined as:

$$ $\Delta^{(t)}=\max(\mathrm{Var}[\partial E^{(t)}]-S^{(t-1)},0),$ $$

where $\mathrm{Var}[\partial E^{(t)}]$ denotes the gradient variance in the $t$-th iteration, and $S^{(t-1)}=\sum_{t_{i}=1}^{t-1}\Delta^{(t_{i})}\cdot I^{(t_{i})}$ denotes the maximum observed gradient variance in the past iterations, where $I^{(t_{i})}$ represents an indicator function $\mathbf{1}\bigl(\Delta^{(t_{i})}\geq\nicefrac{{1}}{{(poly(N,L)K)}}\bigr)$ given a condition inside.

###### Assumption 1 (Bounded Maximum Gradient) .

We assume there exists a positive constant $B_{\partial E}>0$, s.t. the maximum possible gradient $\partial E_{max}$ during qnn’s training satisfies:

$$ $\bigl|\partial E_{max}\bigr|\leq B_{\partial E}.$ $$

Without loss of generality, let’s say $\partial E_{max}\in[-\frac{B_{\partial E}}{2},\frac{B_{\partial E}}{2}]$.

###### Lemma 2 (Boundedness of Gradient Variance) .

Given a certain-size quantum neural network (qnn), the variance of its gradient at the beginning of training, $\mathrm{Var}[\partial E]$, is bounded by:

$$ $\mathrm{Var}[\partial E]\leq(\partial E_{\max}-\partial E_{\min})^{2},$ $$

where $\partial E_{\max}$ and $\partial E_{\min}$ denote the maximum and minimum values of the gradient $\partial E$, respectively.

###### Lemma 3 (Boundedness of EI) .

From Def. [2](https://arxiv.org/html/2502.13166v2#Thmdefinition2) and Lem. [2](https://arxiv.org/html/2502.13166v2#Thmlemma2), in the process of generating initial model parameters $\bm{\theta}_{0}$ for a certain-size qnn, for $\forall$ $t\in\mathbb{Z}^{+}$, there exists a bound for the expected improvement (EI) s.t.

$$ $\Delta^{(t)}\leq(\partial E_{max}-\partial E_{min})^{2}.$ $$

These results indicate that $S^{(t)}$ is $L^{1}$-bounded and integrable for each $t$.
Building upon these lemmas, we investigate the submartingale property and rigorously prove in Lem. [4](https://arxiv.org/html/2502.13166v2#Thmlemma4) that $S^{(t)}$ is a submartingale.

###### Lemma 4 (Informal Statement of Submartingale Property) .

Let $\{I^{(t)}\}_{t\geq 1}$ be an i.i.d. sequence of Bernoulli random variables on a probability space $(\Omega,\mathcal{F},P)$ s.t. $I^{(t)}=1$ with a real number $p\in(0,1]$. Then, $\{S^{(t)}\}_{t\geq 1}$ is a submartingale with respect to the filtration $\{\mathcal{F}^{(t)}\}_{t\geq 1}$ which denotes the collections of all possible events up to time $t$.

Note that the random variable is defined in relation to the comparison between the values of $\Delta^{(t)}$ and $\nicefrac{{1}}{{(poly(N,L)K)}}$. From the definition, it is easy to see that $\Delta^{(t)}\geq\nicefrac{{1}}{{(poly(N,L)K)}}$ when $I^{(t)}=1$, and $\Delta^{(t)}<\nicefrac{{1}}{{(poly(N,L)K)}}$ when $I^{(t)}=0$. More precisely, each $\Delta\cdot I$ (omitting the superscripts) defines a joint distribution of one discrete random variable and one continuous random variable. More details about this intuition are provided in the proofs (Appendix [A](https://arxiv.org/html/2502.13166v2#A1)).

Leveraging the convergence of submartingales and the monotonicity of $S^{(t)}$, we establish in Lem. [5](https://arxiv.org/html/2502.13166v2#Thmlemma5) that $S^{(t)}$ has a supremum, which indicates that our proposed framework can eventually generate effective initial model parameters for qnns that can yield non-negligible gradient variance.

###### Lemma 5 (Boundedness of Submartingale) .

Let $\{S^{(t)}\}_{t\geq 1}$ be a submartingale w.r.t. a $\{\mathcal{F}^{(t)}\}_{t\geq 1}$ s.t. $\sup_{t}\mathbb{E}[|S^{(t)}|]<\infty$. Then, $\{S^{(t)}\}_{t\geq 1}$ is almost surely bounded by a finite constant $B_{S}$ s.t. $S^{(t)}\leq B_{S},\quad\text{a.s.,}\quad\forall\ t\in\mathbb{Z}^{+}.$

Building upon Lem. [5](https://arxiv.org/html/2502.13166v2#Thmlemma5), Thm. [4](https://arxiv.org/html/2502.13166v2#Thmtheorem4) shows the expected hitting time $\mathbb{E}[T_{b}]$ of a bounded submartingale, ensuring that our framework will converge to a desired solution within a finite number of iterations.

###### Theorem 4 (Expected Hitting Time of a Bounded Submartingale) .

Let $\delta=\nicefrac{{p}}{{(poly(N,L)K)}}>0$ with $p$ defined in Lem. [4](https://arxiv.org/html/2502.13166v2#Thmlemma4). Let $T_{b}$ be the hitting time of a bounded submartingale $\{S^{(t)}\}_{t\geq 1}$, where $S^{(t)}\leq B_{S}$ almost surely, for $\forall\ t\in\mathbb{Z}^{+}$ (by Lem. [5](https://arxiv.org/html/2502.13166v2#Thmlemma5)). We define the hitting time as:
$T_{b}=\inf\left\{t\in\mathbb{Z}^{+}:S^{(t)}=b\right\},$
for some threshold $b\leq B_{S}$ such that the set is non-empty almost surely. Then the expected hitting time satisfies:

$$ $\mathbb{E}[T_{b}]\leq\frac{b}{\delta}=\frac{b\cdot poly(N,L)\cdot K}{p}.$ $$

With this theorem, it is straightforward to derive the results for two meaningful cases: (i) $b=\nicefrac{{1}}{{poly(N,L)}}$; and (ii) $b=B_{S}$. This is summarized as Cor. [1](https://arxiv.org/html/2502.13166v2#Thmcorollary1) in the Appendix [A](https://arxiv.org/html/2502.13166v2#A1).
(i) When the threshold $b=\nicefrac{{1}}{{poly(N,L)}}$, the expected hitting time satisfies $\mathbb{E}[T_{b}]\leq\nicefrac{{K}}{{p}}$, where the probability $p$ is defined in Lem. [4](https://arxiv.org/html/2502.13166v2#Thmlemma4). Note that $K$ determines the desired threshold on increment $\Delta^{(t)}$. This is related to the generative power provided by the input GenAI model $f$, which is manifested in Lem. [4](https://arxiv.org/html/2502.13166v2#Thmlemma4). Concretely, if the model $f$ can help return a wanted $\Delta^{(t)}$ with greater $\nicefrac{{1}}{{(poly(N,L)K)}}$ (i.e. smaller $K$) and better probability (i.e. $p$), then the expected stopping iteration $T$ is getting smaller.
(ii) When $b=B_{S}$, i.e., the supremum of the submartingale, $\mathbb{E}[T_{b}]\leq\nicefrac{{B_{S}\cdot poly(N,L)\cdot K}}{{p}}$. This suggests that even when iteratively generating more effective initial parameters for qnns (can yield higher gradient variance), our framework can still converge within a tractable (polynomial) number of iterations if $B_{s}=\mathcal{O}(poly(N,L))$.
Both cases demonstrate that our framework is not only theoretically grounded but also robust in generating meaningful initializations for qnns under different optimization goals.

Furthermore, we discuss extreme cases when $p$ is negligible, e.g., $p=\mathcal{O}(2^{-N})$. One such case involves ansatz-induced BPs Holmes et al. (2022), where initialization-based methods fail in this scenario. Another case arises when llms repeatedly generate identical ineffective outputs. The former case could be addressed by modifying the qnn architecture Holmes et al. (2022), while the latter one can be solved by adjusting hyperparameters, like temperature or top-p, to enhance output diversity Achiam et al. (2023).

###### Definition 2 (Expected Improvement) .

###### Assumption 1 (Bounded Maximum Gradient) .

###### Lemma 2 (Boundedness of Gradient Variance) .

###### Lemma 3 (Boundedness of EI) .

###### Lemma 4 (Informal Statement of Submartingale Property) .

###### Lemma 5 (Boundedness of Submartingale) .

###### Theorem 4 (Expected Hitting Time of a Bounded Submartingale) .

## 5 Experiment

In this section, we first introduce the experimental settings and further present our results in detail.

### Experimental settings.

We evaluate our proposed method across four public datasets, Iris, Wine, Titanic, and MNIST.
We present the dataset statistics and settings in the Appendix [B](https://arxiv.org/html/2502.13166v2#A2).
In the experiment, we analyze the trend of gradient variance by varying the number of qubits, ranging from 2 to 20 in increments of 2 (fixed 2 layers), and the number of layers, spanning from 4 to 40 in steps of 4 (fixed 2 qubits). To obtain reliable results, we repeat the experiments five times and present them as curves (mean) with their bandwidth (standard deviation).
Overall, our framework can generate effective model parameters within 50 iterations. In each iteration, we employ an Adam optimizer with a learning rate of 0.01 and a batch size of 20 to train a qnn with 30 epochs and compute the gradient variance. After training, we compute the expected improvement (EI) and compare it with an assumed lower bound, $\nicefrac{{1}}{{(poly(N,L)K)}}$, in each iteration, where we set $K=T$ in the experiments. We empirically choose the lower bound by $\nicefrac{{1}}{{(T\cdot N^{6})}}$ (Appendix [B](https://arxiv.org/html/2502.13166v2#A2)) and apply it for all cases, as we observe in Fig. [3](https://arxiv.org/html/2502.13166v2#S5.F3) that the magnitudes of gradient variances are comparable across all datasets.
For evaluation, we follow McClean et al. (2018) to assess BP by measuring the gradient variance during the early stage of qnns’ training. A higher gradient variance implies a reduced likelihood of encountering BPs.

Figure: (a) Iris
Refer to caption: x3.png

Figure: Figure 4: Example of three classic distributions commonly used for initialization. In the figure, the red dots represent the initial values of the model parameters.
Refer to caption: x11.png

### Generating initial model parameters of qnn s using our framework can help mitigate BPs.

We analyze gradient variance trends in the first element of qnns’ model parameters across varying qubit and layer settings for three classic initialization distributions, uniform, normal, and beta distributions, which are presented in Fig. [4](https://arxiv.org/html/2502.13166v2#S5.F4) as examples. For each initialization with classic distribution, we compare it (“Classic”) with our proposed methods (“Ours”).
As presented in Fig. [3](https://arxiv.org/html/2502.13166v2#S5.F3), we observe that in the case of using classic initialization, the gradient variance of qnns will significantly decrease as the number of qubits or layers increases. Compared with it, our method can maintain higher variances, indicating that our framework can mitigate BPs better.
In the rest of the experiments, if there is no specific state, we adopt a uniform distribution as prior knowledge for posterior refinement.
Besides the above comparison, we further investigate the contribution of llms in the Appendix [B](https://arxiv.org/html/2502.13166v2#A2).

**Table 1: Comparison of initial model parameters’ generation by accuracy (Acc.) using GPT Hurst et al. (2024), Gemini Team et al. (2024), Claude Anthropic (2024), and LLaMA Grattafiori et al. (2024). Suffix ‘K’ denotes ‘thousand’.**
| llms | Acc. | Max i/o |
| --- | --- | --- |
| GPT-4o | 100% | 128K / 4K |
| GPT-4o Mini | 85% | 128K / 16K |
| Gemini 1.5 Flash | 75% | 1M / 8K |
| Gemini 1.5 Pro | 90% | 2M / 8K |
| Claude 3.5 Sonnet | 100% | 200K / 8K |
| LLaMA 3 70B Instruct | 0% | 8K / 2K |
| LLaMA 3 405B Instruct | 50% | 128K / 2K |

### Comparison of generative performance using llm s.

In our framework, the initial model parameters of qnns are generated by llms. We compare the generative performance under varying qnn structures, such as different numbers of qubits or layers. Specifically, we primarily evaluate whether the correct size of model parameters can be generated by testing 20 combinations in accuracy, fixing either 2 layers while varying qubits from 2 to 20, or 2 qubits while varying layers from 4 to 40. As shown in Tab. [1](https://arxiv.org/html/2502.13166v2#S5.T1), the results indicate that both GPT-4o and Claude 3.5 Sonnet can achieve 100% accuracy in generating the correct shapes of model parameters. Considering 4K output tokens are sufficient for our settings, we mainly use GPT-4o as the backbone llms. We provide additional observation in the Appendix [B](https://arxiv.org/html/2502.13166v2#A2).

Figure: (a) Iris
Refer to caption: x12.png

### Investigation of prompts.

We examine whether the content of prompts influences generative performance. In the experiments, we tested four prompting scenarios: (i) Including both data description and gradient feedback in prompts (Both desc. and feedback), (ii) Including gradient feedback only (No desc.), (iii) Including data description only (No feedback), (iv) Including neither data description nor gradient feedback (Neither desc. nor feedback). As the results presented in Fig. [5](https://arxiv.org/html/2502.13166v2#S5.F5), we observe that suppressing either dataset description or gradient feedback in the prompts leads to a reduction in the gradient variance of qnns. Notably, the reduction is more significant in most cases when gradient feedback is muted compared to the dataset description, suggesting that both factors play a crucial role in mitigating BPs, with gradient feedback contributing significantly more.

Figure: Figure 6: Comparison between two strategies and our framework, which is initialized with the corresponding data distribution for a fair comparison.
Refer to caption: x20.png

### Comparison with initialization-based strategies.

We compare our framework with two representative initialization-based strategies, GaInit Zhang et al. (2022) and BeInit Kulshrestha & Safro (2022). Both of them leverage well-designed Normal and Beta distributions to initialize the qnns, respectively. For a fair comparison, we initialize the qnns with the corresponding distribution. We present the results on Iris in Fig. [6](https://arxiv.org/html/2502.13166v2#S5.F6) as an example. The results demonstrate that our framework can generate initial model parameters of qnns that achieve higher gradient variance at the beginning of training as the model size increases, indicating better mitigation for BPs.

### Analysis of the expected improvement.

We analyze the patterns on the expected improvement (EI) and the corresponding gradient variance across various qnn structures as iterations progress. Representative experiments conducted on Iris are illustrated in Fig. [11](https://arxiv.org/html/2502.13166v2#A2.F11) as an example (Appendix [B](https://arxiv.org/html/2502.13166v2#A2)). Our findings show that the framework can reliably discover meaningful initial parameters regardless of model size. Besides, as the model size grows, more iterations are required to obtain effective initial parameters that enable qnns to maintain higher gradient variance. This is expected, as larger models expand the candidate space, demanding greater computational resources to explore effectively. Both observations verify the Cor. [1](https://arxiv.org/html/2502.13166v2#Thmcorollary1) (Appendix [A](https://arxiv.org/html/2502.13166v2#A1)).

Figure: (a) Iris
Refer to caption: x22.png

### Sensitivity analysis of hyperparameters.

We analyze the sensitivity of hyperparameters, including Temperature and Top P, for llms. Temperature controls the randomness of predictions, with higher values generating more diverse outputs, while Top P affects the probabilities of token selections, balancing generation diversity and structural consistency. To identify optimal settings, we first narrowed down the ranges through manual tuning and then applied grid search to determine the best combinations (Temperature, Top P) for each dataset: Iris (0.5, 0.9), Wine (0.1, 0.45), Titanic (0.8, 0.75), and MNIST (0.8, 0.8), as presented in Fig. [7](https://arxiv.org/html/2502.13166v2#S5.F7). The combinations of the above hyperparameters were used in this study.

Due to the page limit, we present our supplementary results, such as simulation time of qnns, and the computational trade-off, in the Appendix [B](https://arxiv.org/html/2502.13166v2#A2).

## 6 Conclusion

In this study, we aim to mitigate barren plateaus (BPs) by introducing a new AI-driven submartingale-based framework, namely AdaInit. This framework iteratively generates effective initial parameters using generative models, such as llms, for qnns that yield non-negligible gradient variance, thereby mitigating BPs. Our theoretical analysis establishes the submartingale property for the iterative process, ensuring the effective generation. Through extensive experiments across various model scales, we demonstrated that AdaInit outperforms conventional classic initialization methods in maintaining higher gradient variance as qnn’s sizes increase. Overall, this study might initiate a new avenue to explore how llms help mitigate BPs.

### Limitations, future work, and broad impact.

First, our theoretical analyses assume that the maximum gradient of qnns is bounded by a positive constant, implying that gradient explosion does not occur during training, a condition that is typically satisfied in practice. Second, due to the practical limitations of quantum simulation, our experiments are constrained to qnns with up to 20 qubits. We also assume an idealized setting where quantum measurements are noise-free. Moreover, our current scope excludes ansatz-induced BPs, which may be mitigated through architectural modifications, as discussed above.
For future work, we plan to (i) accelerate the convergence of the iterative process and (ii) expand the applicability of our framework beyond BP mitigation. In particular, it can be leveraged to guide qnn architecture design or to identify optimal model parameters in training.
More broadly, our framework can support robust qnn training across various domains, such as healthcare, where robustness and reliability are critical.

Ethics Statement
This work does not involve human subjects, sensitive personal data, or tasks with foreseeable negative societal impact. The datasets used (Iris, Wine, Titanic, and MNIST) are standard, publicly available benchmarks. We ensured compliance with their respective licenses and data usage guidelines. The proposed framework is designed to mitigate barren plateaus and does not directly enable harmful applications. Nonetheless, as with other advances in optimization, the method could potentially be applied in sensitive domains; in such cases, practitioners should carefully consider fairness, privacy, and security concerns in line with the ICLR Code of Ethics.

Reproducibility Statement
We have made every effort to ensure reproducibility of our results. Detailed theoretical proofs are included in Appendix A, whereas experimental settings, dataset splits, hyperparameters, architecture of the backbone quantum circuit, and the computing infrastructure are described in Appendix B. Besides, we provide sample code and datasets in the supplementary materials, which will be made publicly available upon publication. These resources, together with the descriptions of prompts, model parameters, and evaluation protocols, are intended to enable independent verification and extension of our work.

## References

- Achiam et al. (2023)
Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al.
Gpt-4 technical report.
*arXiv preprint arXiv:2303.08774*, 2023.
- Anthropic (2024)
Anthropic.
Introducing claude 3.5 sonnet, June 2024.
URL [https://www.anthropic.com/news/claude-3-5-sonnet](https://www.anthropic.com/news/claude-3-5-sonnet).
Accessed: 2025-02-15.
- Bharti & Haug (2021)
Kishor Bharti and Tobias Haug.
Quantum-assisted simulator.
*Physical Review A*, 104(4):042418, 2021.
- Cerezo et al. (2021)
Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles.
Cost function dependent barren plateaus in shallow parametrized quantum circuits.
*Nature communications*, 12(1):1791, 2021.
- Cunningham & Zhuang (2024)
Jack Cunningham and Jun Zhuang.
Investigating and mitigating barren plateaus in variational quantum circuits: A survey.
*arXiv preprint arXiv:2407.17706*, 2024.
- Du et al. (2022)
Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, and Dacheng Tao.
Quantum circuit architecture search for variational quantum algorithms.
*npj Quantum Information*, 8(1):62, 2022.
- Farhi et al. (2014)
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann.
A quantum approximate optimization algorithm.
*arXiv preprint arXiv:1411.4028*, 2014.
- Freeman & Stephenson (2025)
Nic Freeman and Robin Stephenson.
Mas352: Stochastic processes and financial mathematics (notes).
[https://nicfreeman1209.github.io/Website/MASx52/html/notes_1.html](https://nicfreeman1209.github.io/Website/MASx52/html/notes_1.html), 2025.
Accessed: 2025-05-15.
- Grant et al. (2019)
Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti.
An initialization strategy for addressing barren plateaus in parametrized quantum circuits.
*Quantum*, 2019.
- Grattafiori et al. (2024)
Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al.
The llama 3 herd of models.
*arXiv preprint arXiv:2407.21783*, 2024.
- Grimsley et al. (2023)
Harper R Grimsley, George S Barron, Edwin Barnes, Sophia E Economou, and Nicholas J Mayhall.
Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus.
*npj Quantum Information*, 9(1):19, 2023.
- Heyraud et al. (2023)
Valentin Heyraud, Zejian Li, Kaelan Donatella, Alexandre Le Boité, and Cristiano Ciuti.
Efficient estimation of trainability for variational quantum circuits.
*PRX Quantum*, 4(4):040335, 2023.
- Holmes et al. (2022)
Zoë Holmes, Kunal Sharma, Marco Cerezo, and Patrick J Coles.
Connecting ansatz expressibility to gradient magnitudes and barren plateaus.
*PRX quantum*, 3(1):010313, 2022.
- Hurst et al. (2024)
Aaron Hurst, Adam Lerer, Adam P Goucher, Adam Perelman, Aditya Ramesh, Aidan Clark, AJ Ostrow, Akila Welihinda, Alan Hayes, Alec Radford, et al.
Gpt-4o system card.
*arXiv preprint arXiv:2410.21276*, 2024.
- Kandala et al. (2017)
Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta.
Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.
*nature*, 549(7671):242–246, 2017.
- Kashif & Al-Kuwari (2024)
Muhammad Kashif and Saif Al-Kuwari.
Resqnets: a residual approach for mitigating barren plateaus in quantum neural networks.
*EPJ Quantum Technology*, 2024.
- Kashif et al. (2024)
Muhammad Kashif, Muhammad Rashid, Saif Al-Kuwari, and Muhammad Shafique.
Alleviating barren plateaus in parameterized quantum machine learning circuits: Investigating advanced parameter initialization strategies.
In *2024 Design, Automation & Test in Europe Conference & Exhibition (DATE)*, pp. 1–6. IEEE, 2024.
- Kulshrestha & Safro (2022)
Ankit Kulshrestha and Ilya Safro.
Beinit: Avoiding barren plateaus in variational quantum algorithms.
In *2022 IEEE international conference on quantum computing and engineering (QCE)*, pp. 197–203. IEEE, 2022.
- Li et al. (2021)
Guangxi Li, Zhixin Song, and Xin Wang.
Vsql: Variational shadow quantum learning for classification.
In *Proceedings of the AAAI conference on artificial intelligence*, 2021.
- Liu et al. (2023)
Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, Yong-Jian Han, and Guo-Ping Guo.
Mitigating barren plateaus with transfer-learning-inspired parameter initializations.
*New Journal of Physics*, 25(1):013039, 2023.
- Liu et al. (2024)
Xia Liu, Geng Liu, Hao-Kai Zhang, Jiaxin Huang, and Xin Wang.
Mitigating barren plateaus of variational quantum eigensolvers.
*IEEE Transactions on Quantum Engineering*, 2024.
- McClean et al. (2018)
Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven.
Barren plateaus in quantum neural network training landscapes.
*Nature communications*, 9(1):4812, 2018.
- Mele et al. (2022)
Antonio A Mele, Glen B Mbeng, Giuseppe E Santoro, Mario Collura, and Pietro Torta.
Avoiding barren plateaus via transferability of smooth solutions in a hamiltonian variational ansatz.
*Physical Review A*, 106(6):L060401, 2022.
- Ortiz Marrero et al. (2021)
Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe.
Entanglement-induced barren plateaus.
*PRX quantum*, 2(4):040316, 2021.
- Ostaszewski et al. (2021)
Mateusz Ostaszewski, Edward Grant, and Marcello Benedetti.
Structure optimization for parameterized quantum circuits.
*Quantum*, 5:391, 2021.
- Park & Killoran (2024)
Chae-Yeun Park and Nathan Killoran.
Hamiltonian variational ansatz without barren plateaus.
*Quantum*, 8:1239, 2024.
- Preskill (2018)
John Preskill.
Quantum computing in the nisq era and beyond.
*Quantum*, 2:79, 2018.
- Qi et al. (2023)
Han Qi, Lei Wang, Hongsheng Zhu, Abdullah Gani, and Changqing Gong.
The barren plateaus of quantum neural networks: review, taxonomy and trends.
*Quantum Information Processing*, 22(12):435, 2023.
- Rappaport et al. (2023)
Sonny Rappaport, Gaurav Gyawali, Tiago Sereno, and Michael J Lawler.
Measurement-induced landscape transitions in hybrid variational quantum circuits.
*arXiv preprint arXiv:2312.09135*, 2023.
- Sack et al. (2022)
Stefan H Sack, Raimel A Medina, Alexios A Michailidis, Richard Kueng, and Maksym Serbyn.
Avoiding barren plateaus using classical shadows.
*PRX Quantum*, 3(2):020365, 2022.
- Sannia et al. (2024)
Antonio Sannia, Francesco Tacchino, Ivano Tavernelli, Gian Luca Giorgi, and Roberta Zambrini.
Engineered dissipation to mitigate barren plateaus.
*npj Quantum Information*, 10(1):81, 2024.
- Selvarajan et al. (2023)
Raja Selvarajan, Manas Sajjan, Travis S Humble, and Sabre Kais.
Dimensionality reduction with variational encoders based on subsystem purification.
*Mathematics*, 2023.
- Shang & Shi (2025)
Yun Shang and Xiao Shi.
Avoiding barren plateaus via gaussian mixture model.
*New Journal of Physics*, 2025.
- Stein et al. (2021)
Samuel A Stein, Betis Baheri, Daniel Chen, Ying Mao, Qiang Guan, Ang Li, Bo Fang, and Shuai Xu.
Qugan: A quantum state fidelity based generative adversarial network.
In *2021 IEEE International Conference on Quantum Computing and Engineering (QCE)*, pp. 71–81. IEEE, 2021.
- Suzuki et al. (2021)
Yudai Suzuki, Hiroshi Yano, Rudy Raymond, and Naoki Yamamoto.
Normalized gradient descent for variational quantum algorithms.
In *2021 IEEE International Conference on Quantum Computing and Engineering (QCE)*, pp. 1–9. IEEE, 2021.
- Team et al. (2024)
Gemini Team, Petko Georgiev, Ving Ian Lei, Ryan Burnell, Libin Bai, Anmol Gulati, Garrett Tanzer, Damien Vincent, Zhufeng Pan, Shibo Wang, et al.
Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context.
*arXiv preprint arXiv:2403.05530*, 2024.
- Tüysüz et al. (2023)
Cenk Tüysüz, Giuseppe Clemente, Arianna Crippa, Tobias Hartung, Stefan Kühn, and Karl Jansen.
Classical splitting of parametrized quantum circuits.
*Quantum Machine Intelligence*, 2023.
- Williams (1991)
David Williams.
*Probability with martingales*.
Cambridge university press, 1991.
- Zhang et al. (2022)
Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao.
Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits.
*Advances in Neural Information Processing Systems*, 2022.
- Zhuang et al. (2024)
Jun Zhuang, Jack Cunningham, and Chaowen Guan.
Improving trainability of variational quantum circuits via regularization strategies.
*arXiv preprint arXiv:2405.01606*, 2024.

## Appendix A Proofs

In this section, we provide formal proofs that consolidate the theoretical guarantees of our framework.

###### Proof for Lemma 2 .

We denote a sequence of gradient $\bm{\partial E}=\{\partial E^{(t)}\}_{t=0}^{T_{tr}}$, where $T_{tr}$ represents the number of training epochs for a qnn. Within this sequence, we denote $\partial E_{max}$, $\partial E_{min}$, and $\overline{\partial E}$ as the maximum, minimum, and mean values of the gradient. For $\forall$ $t\in\mathbb{Z}^{+}$, we have:

$$ $\partial E^{(t)},\overline{\partial E}\in[\partial E_{min},\partial E_{max}],$ $$

then the gap between $\partial E^{(t)}$ and $\overline{\partial E}$ will not exceed the range of $[\partial E_{min},\partial E_{max}]$:

$$ $|\partial E^{(t)}-\overline{\partial E}|\leq\partial E_{max}-\partial E_{min}.$ $$

Thus, we have:

$$ $\displaystyle\mathrm{Var}[\partial E]$ $\displaystyle=\frac{1}{T_{tr}}\sum_{t=1}^{T_{tr}}(\partial E^{(t)}-\overline{\partial E})^{2}$ $\displaystyle\leq\frac{1}{T_{tr}}\sum_{t=1}^{T_{tr}}(\partial E_{max}-\partial E_{min})^{2}$ $\displaystyle=(\partial E_{max}-\partial E_{min})^{2}.$ $$

Thus, the gradient variance $\mathrm{Var}[\partial E]$ satisfies the bound $\mathrm{Var}[\partial E]\leq(\partial E_{max}-\partial E_{min})^{2}$.
∎

###### Proof for Lemma 3 .

From Def. [2](https://arxiv.org/html/2502.13166v2#Thmdefinition2), for $\forall$ $t\in\mathbb{Z}^{+}$, in the $t$-th search iteration, we have:

$$ $\Delta^{(t)}=\max(\mathrm{Var}[\partial E^{(t)}]-S^{(t-1)},0).$ $$

Combining with Lem. [2](https://arxiv.org/html/2502.13166v2#Thmlemma2), for $\forall$ $t\in\mathbb{Z}^{+}$, we have:

$$ $\mathrm{Var}[\partial E^{(t)}],S^{(t-1)}\leq(\partial E_{max}-\partial E_{min})^{2}.$ $$

The above equation holds true as $S^{(t-1)}$ denotes the historical maximum gradient variance in the past iterations. Thus, we have:

$$ $\mathrm{Var}[\partial E^{(t)}]-S^{(t-1)}\leq(\partial E_{max}-\partial E_{min})^{2},$ $$

which indicates that:

$$ $\Delta^{(t)}\leq(\partial E_{max}-\partial E_{min})^{2}.$ $$

∎

Before introducing the formal statement of submartingale property, we first define the random variables. Let $\alpha=\nicefrac{{1}}{{(poly(N,L)K)}}$.
To avoid confusion, notably, we clarify that the $K$ refers to a previously defined parameter, whereas in Tab. [1](https://arxiv.org/html/2502.13166v2#S5.T1), the suffix ‘K’ attached to numbers represents the unit ‘thousand’.
Formally, we define discrete random variables $I^{(t)}$ and continuous random variables $\Delta^{{(t)}}$ as follows:

$$ $\begin{cases}P(I^{(t)}=1)=P(\Delta^{(t)}\geq\alpha)=p,\\ P(I^{(t)}=0)=P(\Delta^{(t)}<\alpha)=1-p,\end{cases}$ (5) $$

with a real number $p\in(0,1]$. Hence, $I^{(t)}$ is a Bernoulli random variable as an indicator of the event $\Delta^{(t)}\geq\alpha$, and its associated continuous random variable $\Delta^{{(t)}}$ is implicitly defined with an arbitrary probability density function $p(y)$ that satisfies

$$ $P(\Delta^{(t)}\geq\alpha)=\int_{\alpha}^{\infty}p(y)dy=p.$ (6) $$

With such a relationship between them, the following can be easily verified, for all $t\in\mathbb{Z}^{+}$,

$$ $P(I^{(t)}=x|\Delta^{(t)}=y)=\begin{cases}x,&y\geq\alpha\\ 1-x,&\hbox{o.w.}\end{cases}.$ (7) $$

###### Lemma 6 (Submartingale Property, formal statement of Lemma 4 ) .

Let $\{I^{(t)},\Delta^{(t)}\}_{t\geq 1}$ be a sequence of i.i.d. random variables as defined in [Eq. 5](https://arxiv.org/html/2502.13166v2#A1.E5). We define joint random variables $W^{(t)}=\Delta^{(t)}\cdot I^{(t)}$ and the natural filtration $\mathcal{F}^{(t)}=\sigma\bigl(W^{(1)},\cdots,W^{(t)}\bigr)$. Then the sequence $\{S^{(t)}\}_{t\geq 1}$ as defined by Def. [2](https://arxiv.org/html/2502.13166v2#Thmdefinition2) is a submartingale with respect to the filtration $\{\mathcal{F}^{(t)}\}_{t\geq 1}$.

###### Proof for Lemma 6 .

According to Def. [1](https://arxiv.org/html/2502.13166v2#Thmdefinition1), a process $S^{(t)}$ is a submartingale relative to $(\Omega,\mathcal{F},P)$ if it satisfies Adaptedness, Integrability, and Submartingale.

###### Proof for Lemma 2 .

###### Proof for Lemma 3 .

###### Lemma 6 (Submartingale Property, formal statement of Lemma 4 ) .

###### Proof for Lemma 6 .

### Adaptedness.

We first aim to verify that $S^{(t)}$ is determined based on the information available up to past $t$ iterations. By Def. [2](https://arxiv.org/html/2502.13166v2#Thmdefinition2), $S^{(t)}=\sum_{t_{i}=1}^{t}\Delta^{(t_{i})}\cdot I^{(t_{i})}=\sum_{t_{i}=1}^{t}W^{(t_{i})}$ is a finite sum of random variables that are measurable w.r.t. $\sigma\bigl(W^{(1)},\dots,W^{(t)}\bigr)$ (or $\sigma\bigl(I^{(1)},\dots,I^{(t)}\bigr)$ for short due to the relationship between $I^{(t)}$ and $\Delta^{(t)}$ as shown in [Eq. 5](https://arxiv.org/html/2502.13166v2#A1.E5)). Thus, $S^{(t)}$ is also measurable w.r.t. $\mathcal{F}^{(t)}$, ensuring the adaptedness.

### Integrability.

In Lem. [3](https://arxiv.org/html/2502.13166v2#S4.Ex4), $\Delta^{(t)}\leq(\partial E_{max}-\partial E_{min})^{2}$ for $\forall$ $t\in\mathbb{Z}^{+}$. Thus,

$$ $\displaystyle\mathbb{E}[|S^{(t)}|]$ $\displaystyle=\mathbb{E}\Bigl[\Bigl|\sum_{t_{i}=1}^{t}\Delta^{(t_{i})}\cdot I^{(t_{i})}\Bigr|\Bigr]$ $\displaystyle\leq\mathbb{E}\Bigl[\Bigl|\sum_{t_{i}=1}^{t}(\partial E_{max}-\partial E_{min})^{2}\cdot I^{(t_{i})}\Bigr|\Bigr]$ $\displaystyle<\infty,$ $$

which ensures $\mathbb{E}[|S^{(t)}|]$ is integrable for each $t$.

### Submartingale.

Before proving this condition, we show the following necessary inequality: with $\alpha=\nicefrac{{1}}{{(poly(N,L)K)}}$,

$$ $\displaystyle\mathbb{E}[\Delta^{(t)}\cdot I^{(t)}]$ $\displaystyle=\sum_{x=0,1}\int_{-\infty}^{\infty}P(I^{(t)}=x|\Delta^{(t)}=y)\cdot p(y)\cdot x\cdot y\ dy$ $\displaystyle=\int_{\alpha}^{\infty}p(y)\cdot y\ \ dy$ $\displaystyle\geq\alpha\int_{\alpha}^{\infty}p(y)dy$ $\displaystyle=\alpha p\ ,$ (8) $$

where the second step follows from [Eq. 7](https://arxiv.org/html/2502.13166v2#A1.E7) and the last step uses [Eq. 6](https://arxiv.org/html/2502.13166v2#A1.E6). Apparently $\alpha p>0$.

We observe that

$$ $S^{(t)}=S^{(t-1)}+\Delta^{(t)}\cdot I^{(t)}.$ $$

Since $S^{(t-1)}$ is $\mathcal{F}^{(t-1)}$-measurable and $\{\Delta^{(t)}\}_{t\geq 1}$ is i.i.d., thus,

$$ $\displaystyle\mathbb{E}\bigl[S^{(t)}\big|\mathcal{F}^{(t-1)}\bigr]$ $\displaystyle=\mathbb{E}\bigl[S^{(t-1)}+\Delta^{(t)}\cdot I^{(t)}\big|\mathcal{F}^{(t-1)}\bigr]$ $\displaystyle=S^{(t-1)}+\mathbb{E}[\Delta^{(t)}\cdot I^{(t)}]$ $\displaystyle\geq S^{(t-1)}+\alpha p$ $\displaystyle\geq S^{(t-1)},$ $$

where the last two step applies [Appendix A](https://arxiv.org/html/2502.13166v2#A1.Ex16).

Thus, the submartingale condition holds true for $\forall$ $t\geq 1$ s.t.

$$ $\mathbb{E}\bigl[S^{(t)}\big|\mathcal{F}^{(t-1)}\bigr]\geq S^{(t-1)},\quad\forall\ t\geq 1.$ $$

∎
Intuitively, $S^{(t)}$ tracks the cumulative amount of non-trivial variance improvements observed over the iterations — akin to measuring meaningful progress in exploration.

###### Proof for Lemma 5 .

Since the process $\{S^{(t)}\}_{t\geq 1}$ is a $L^{1}$-bounded submartingale s.t. $\sup_{t}\mathbb{E}[|S^{(t)}|]<\infty$, we apply the Doob’s Forward Convergence Theorem (by Thm. [1](https://arxiv.org/html/2502.13166v2#Thmtheorem1)), which guarantees the almost sure existence of a finite random variable $S^{(\infty)}$ s.t. $S^{(\infty)}=\lim_{t\to\infty}S^{(t)}$. This implies that the process $\{S^{(t)}\}$ has a well-defined almost sure limit.

Furthermore, if $\{S^{(t)}\}$ is monotone increasing, i.e., $S^{(t)}\leq S^{(t+1)}$, a.s., $\forall$ $t\in\mathbb{Z}^{+}$, then the limit $S^{(\infty)}$ serves as a supremum for the entire process. By Defining $B_{S}:=\sup_{t}S^{(t)}=S^{(\infty)}$, we obtain a desired bound $S^{(t)}\leq B_{S}$, a.s., $\forall$ $t\in\mathbb{Z}^{+}$.
∎

###### Proof for Theorem 4 .

To analyze the expected hitting time, we first construct a drift-adjusted process $\{Z^{(t)}\}_{t\geq 1}$ adapted to a filtration $\{\mathcal{F}^{(t)}\}_{t\geq 1}$ as $Z^{(t)}=S^{(t)}-\delta t$, where $\delta=\nicefrac{{p}}{{(poly(N,L)K)}}>0$.

Given $\alpha=\nicefrac{{1}}{{(poly(N,L)K)}}$ and follow those similar steps in the proof for Lem. [6](https://arxiv.org/html/2502.13166v2#Thmlemma6), we can derive

$$ $\mathbb{E}[\Delta^{(t)}\cdot I^{(t)}]\geq\alpha p=\delta,$ (9) $$

where the last step is by definition of $\delta$.

We then verify that ${Z^{(t)}}$ is also a submartingale:

• Adaptedness: Similr to $S^{(t)}$, ${Z^{(t)}}$ is also determined by the past $t$ iterations w.r.t. the same filtration $\sigma\bigl(W^{(1)},\dots,W^{(t)}\bigr)$ as $S^{(t)}$. Thus, ${Z^{(t)}}$ can meet the adaptedness.

• Integrability: Given that $S^{(t)}$ is $L^{1}$-bounded, and $Z^{(t)}$ is obtained by subtracting a deterministic finite value $\delta t$ from $S^{(t)}$, it follows immediately that $Z^{(t)}$ is also $L^{1}$-bounded, i.e., $\mathbb{E}[|Z^{(t)}|]$ is integrable for each $t$.

• Submartingale: We further show that $Z^{(t)}$ meets the submartingale inequality as follows.

$$ $\displaystyle\mathbb{E}[Z^{(t+1)}\mid\mathcal{F}^{(t)}]$ $\displaystyle=\mathbb{E}[S^{(t+1)}-\delta(t+1)\mid\mathcal{F}^{(t)}]$ $\displaystyle=\mathbb{E}[S^{(t+1)}\mid\mathcal{F}^{(t)}]-\delta(t+1)$ $\displaystyle=\mathbb{E}[S^{(t)}+\Delta^{(t+1)}\cdot I^{(t+1)}\mid\mathcal{F}^{(t)}]-\delta(t+1)$ $\displaystyle=S^{(t)}+\mathbb{E}[\Delta^{(t+1)}\cdot I^{(t+1)}\mid\mathcal{F}^{(t)}]-\delta(t+1)$ $\displaystyle\geq S^{(t)}+\delta-\delta(t+1)$ $\displaystyle=Z^{(t)},$ $$

where the fifth step is obtained by combining the fact that $\Delta^{(t+1)}\cdot I^{(t+1)}$ is independent of $\mathcal{F}^{(t)}$ and [Eq. 9](https://arxiv.org/html/2502.13166v2#A1.E9).

Second, we define the hitting time $T_{b}$ as $T_{b}=\inf\left\{T\in\mathbb{Z}^{+}:S^{(T)}=b\right\}$. Without loss of generality, we assume $b\leq B_{S}$ since $S^{(t)}\leq B_{S}$ a.s., for $\forall\ t\in\mathbb{Z}^{+}$ (by Lem. [5](https://arxiv.org/html/2502.13166v2#Thmlemma5)). We further verify that $T_{b}$ is a bounded stopping time as follows.

We observe that $\{T_{b}\leq T\}=\{\exists\ t\leq T\hbox{ such that }S^{(t)}=b\}=\bigcup_{t=0}^{T}\{S^{(t)}=b\}$.
Since $S^{(t)}$ is $\mathcal{F}^{(T)}$ measurable for $\forall\ t\leq T$, we have $\{S^{(t)}=b\}\in\mathcal{F}^{(T)}$, which indicates that the finite union $\bigcup_{t=0}^{T}\{S^{(t)}=b\}\in\mathcal{F}^{(T)}$. Hence, $\{T_{b}\leq T\}\in\mathcal{F}^{(T)}$, which by definition shows that $T_{b}$ is a bounded stopping time.

Third, we define $T_{b}\wedge t$ as $\min(T_{b},\ t)$. Given that $T_{b}$ is a bounded stopping time, $T_{b}\wedge t$ is also a bounded stopping time (by Lem. [1](https://arxiv.org/html/2502.13166v2#Thmlemma1)). Based on this condition, the Doob’s Optional Stopping Theorem implies that $\mathbb{E}[Z^{(T_{b}\wedge t)}]\geq\mathbb{E}[Z^{(0)}]=0$ (by Thm. [2](https://arxiv.org/html/2502.13166v2#Thmtheorem2)). Thus, we have:

$$ $\mathbb{E}[S^{(T_{b}\wedge t)}-\delta(T_{b}\wedge t)]\geq 0$ $$

implying

$$ $\mathbb{E}[S^{(T_{b}\wedge t)}]\geq\delta\mathbb{E}[T_{b}\wedge t].$ $$

Since $S^{(t)}$ is non-decreasing and bounded by $B_{S}$, we have $T_{b}\wedge t=T_{b}$ as $t\to\infty$ almost surely, which implies that $\mathbb{E}[T_{b}\wedge t]=\mathbb{E}[T_{b}]$.
Moreover, by the definition of $T_{b}$, it follows that $S^{(T_{b}\wedge t)}\to b$ as $t\to\infty$, i.e., $\mathbb{E}[S^{(T_{b}\wedge t)}]$ is bounded by a dominating constant $b$. So, by the Dominated Convergence Theorem, as $t\to\infty$, we have $\mathbb{E}[S^{(T_{b}\wedge t)}]\to\mathbb{E}[S^{(T_{b})}]=b$ (by Thm. [3](https://arxiv.org/html/2502.13166v2#Thmtheorem3)).

By integrating the above equations and taking the limit, we conclude:

$$ $\mathbb{E}[T_{b}]\leq\frac{b}{\delta}=\frac{bK\cdot poly(N,L)}{p}.$ $$

∎

Thus, the following result can be derived immediately by plugging concrete values of $b$ into Thm. [4](https://arxiv.org/html/2502.13166v2#Thmtheorem4).

###### Corollary 1 (Expected Hitting Time Under Specific Thresholds) .

- 1.
With an expected number of $\nicefrac{{K}}{{p}}$ iterations, Algo. [1](https://arxiv.org/html/2502.13166v2#alg1) can identify a candidate model parameter $\bm{\theta}^{*}_{0}$ that has $\mathrm{Var}[\partial E]\approx\nicefrac{{1}}{{poly(N,L)}}$.
- 2.
With an expected number of $\nicefrac{{B_{S}\cdot K\cdot poly(N,L)}}{{p}}$ iterations, Algo. [1](https://arxiv.org/html/2502.13166v2#alg1) can identify a candidate model parameter $\bm{\theta}^{*}_{0}$ that has $\mathrm{Var}[\partial E]=\mathcal{O}(poly(N,L))$.

###### Proof for Lemma 5 .

###### Proof for Theorem 4 .

###### Corollary 1 (Expected Hitting Time Under Specific Thresholds) .

## Appendix B Supplementary Experiments

In this section, we present supplementary details about our experimental results.

**Table 2: Statistics of datasets. $\left|D\right|$, $\left|F\right|$, and $\left|C\right|$ denote the original number of instances, features, and classes, respectively. “Split” denotes the split instances for the train, validation, and test data.**
| Dataset | $\left|D\right|$ | $\left|F\right|$ | $\left|C\right|$ | Splits |
| --- | --- | --- | --- | --- |
| Iris | 150 | 4 | 3 | 60:20:20 |
| Wine | 178 | 13 | 3 | 80:20:30 |
| Titanic | 891 | 11 | 2 | 320:80:179 |
| MNIST | 60,000 | 784 | 10 | 320:80:400 |

### Dataset.

We evaluate our proposed method across four public datasets that are widely used in quantum machine learning. Iris (^1^11[https://archive.ics.uci.edu/ml/datasets/iris](https://archive.ics.uci.edu/ml/datasets/iris) (Fisher, 1936)) is a classic machine-learning benchmark that measures various attributes of three-species iris flowers. Wine (^2^22[https://archive.ics.uci.edu/ml/datasets/iris](https://archive.ics.uci.edu/ml/datasets/iris) (Fisher, 1936)) is a well-known dataset that includes 13 attributes of chemical composition in wines. Titanic (^3^33[https://www.kaggle.com/c/titanic](https://www.kaggle.com/c/titanic) (Kaggle, 2012)) contains historical data about passengers aboard the Titanic and is typically used to predict survival. MNIST (^4^44[http://yann.lecun.com/exdb/mnist/](http://yann.lecun.com/exdb/mnist/) (LeCun et al., 2010)) is a widely used small benchmark in computer vision. This benchmark consists of 28$\times$28 gray-scale images of handwritten digits from 0 to 9.
We follow the settings of BeInit Kulshrestha & Safro (2022) and conduct experiments in binary classification. Specifically, we randomly sub-sample a certain number of instances from the first two classes of each dataset to create a new subset. After sub-sampling, we employ the t-SNE technique to reduce the feature dimensions to ensure they do not exceed the number of available qubits. The statistics of the original datasets, along with the data splits for training, validation, and testing, are presented in Table [2](https://arxiv.org/html/2502.13166v2#A2.T2). Importantly, the total number of sub-sampled instances corresponds to the sum of the split datasets. For instance, in the Iris dataset, the total number of sub-sampled instances is 100.

Figure: Figure 8: Comparison of model parameter initializations using a classic method, random initializer (RI), and llms. All methods apply a uniform distribution.
Refer to caption: x26.png

### Contribution of llm s.

Besides comparing our framework with the classic method, we further investigate llms’ contribution to the initialization process on Iris. Specifically, we compare the generator within our framework when initialized using a random initializer (RI), which uniformly generates the parameters, versus using llm-based uniform initialization (llms). As shown in Fig. [8](https://arxiv.org/html/2502.13166v2#A2.F8), “RI” performs comparably to, or even worse than, the “Classic” method (uniform), as it generates random parameters without any guided refinement. In contrast, “llms” consistently outperforms both baselines, achieving significantly higher gradient variance. These findings suggest that the llm-driven generator can more effectively explore the parameter space and identify better initializations within a limited number of iterations.

**Table 3: Comparison of generated model parameters (layer, expected shape, and actual shape) between two open-source llms—LLaMA 3 70B Instruct and LLaMA 3 405B Instruct—evaluated under various numbers of qubits ($N$) or layers ($L$).**
| LLaMA 3 | Variables | Layer | Expected | Actual |
| --- | --- | --- | --- | --- |
| 70B | $N\in[2,20]$ | 0 | (2, $N$, 3) | (2, 3) |
| 70B | $L\in[4,40]$ | 0 | ($L$, 2, 3) | ($L$, 3) |
| 405B | $L\in[4,40]$ | 1 | (2, 2) | (2, 2, 2) |

### Observations of open-source models.

We observe that two LLaMA 3 open-source llms Grattafiori et al. (2024) perform significantly worse. To further understand these failures, we present representative cases in Tab. [3](https://arxiv.org/html/2502.13166v2#A2.T3) and observe that both models fail to generate correct shapes of model parameters, indicating that these open-source models struggle to follow precise structural instructions and suggesting their current limitations in shape-constrained generation tasks.

Figure: Figure 9: Assessment of the simulation time in qnn training.
Refer to caption: x27.png

### Simulation time in qnn training.

We assess the simulation time of qnn training under varying model sizes (number of qubits, $N\in[2,20]$) and subsampled MNIST dataset sizes (number of instances, $|D|\in[800,4000]$). We train QNNs for 30 epochs and present the average runtime per epoch. When varying $N$, we fix the number of layers $L$ at 2; when varying $|D|$, we fix both $N$ and $L$ as 2.
As presented in Fig. [9](https://arxiv.org/html/2502.13166v2#A2.F9), in a classical simulated environment, the average training time of qnns increases exponentially w.r.t. the number of qubits, while it grows roughly linearly with the dataset size. These observations reflect an inherent scalability issue in classical simulation of quantum systems. Such limitations are widely acknowledged in the quantum computing community and are unlikely to be fully overcome until practical quantum hardware becomes more accessible.

Figure: Figure 10: Trade-off analysis between computational cost and performance benefits for 2 qubits (left) and 20 qubits (right) setups.
Refer to caption: x29.png

### Trade-off analysis.

To analyze the trade-off in Fig. [11](https://arxiv.org/html/2502.13166v2#A2.F11), we present the relationship between computational cost (measured by the number of search iterations) and performance benefits (quantified by gradient variance) in Fig. [10](https://arxiv.org/html/2502.13166v2#A2.F10).
We observe a roughly linear relationship between them. In the 2-qubit case, 40% cost yields over 60% gain, while in the 20-qubit case, 35% cost yields over 44% gain, showing strong early-stage cost-effectiveness. The diminishing returns after a few iterations align with submartingale optimization behavior, and the consistent trends across scales highlight AdaInit’s suitability for budget-aware scenarios.

Figure: (a) The number of qubits ranges from 2 to 20.
Refer to caption: x31.png

### Patterns of Expected Improvement (EI).

Due to limited pages, we present the patterns of EI in Fig. [11](https://arxiv.org/html/2502.13166v2#A2.F11).

Figure: Figure 12: Trade-off analysis of the assumed lower bound, comparing polynomial terms against the exponential baseline, shown on a log scale.
Refer to caption: x51.png

### Empirical analysis of the assumed lower bound.

To determine the assumed lower bound, $\nicefrac{{1}}{{(poly(N,L)K)}}$, we conduct a trade-off analysis. A larger polynomial coefficient enlarges the admissible regime, but at the cost of including cases with vanishingly small gradient variance, whereas a smaller coefficient may filter out meaningful expected improvements, thereby preventing the framework from effectively exploring initial parameters. To proactively mitigate BPs, we restrict our attention to the range of qubits that are particularly susceptible to BPs. As illustrated in Fig. [12](https://arxiv.org/html/2502.13166v2#A2.F12), we vary the polynomial coefficient and compare against the exponential baseline $\nicefrac{{1}}{{(K2^{2N})}}$. Considering the trade-off, we empirically select $\nicefrac{{1}}{{(KN^{6})}}$ as the lower bound.

### Prompt designs.

Before presenting the prompts, we first introduce the notation for the hyperparameter in the prompts. ‘nlayers’, ‘nqubits’, ‘nrot’, ‘nclasses’ denote the number of layers, qubits, rotation gates, and classes for the qnn, respectively. ‘init’ denotes the initial data distribution for the qnn. ‘data_desc’ denotes the data description. ‘feedback’ denotes the gradient feedback from the previous iteration.

Figure: Figure 13: Architecture of our backbone quantum circuit. The number of rotation gates in this study is fixed as 3.
Refer to caption: x52.png

### Model architecture of the quantum circuit.

In this study, we evaluate our framework using a backbone qnn consisting of a quantum circuit followed by a fully connected layer. Classical data are first encoded into quantum states via angle encoding, where each feature is mapped to rotation gates (e.g., $R_{X}$) on a specific qubit. This encoding maps data into the Hilbert space while preserving differentiability. The circuit applies repeated layers of parameterized rotations ($R_{X}$, $R_{Y}$, $R_{Z}$) and linear-topology CNOT gates for entanglement. After computation, the quantum state is measured in the computational basis, and expectation values of Pauli-Z operators are computed and used as circuit outputs. These values are then processed by the classical fully connected layer. The overall architecture is adaptable in terms of layers, qubits, and rotation gates, as illustrated in Fig. [13](https://arxiv.org/html/2502.13166v2#A2.F13).

### Hardware and software.

The experiment is conducted on a server with the following settings:

- •
Operating System: Ubuntu 22.04.3 LTS
- •
CPU: Intel Xeon w5-3433 @ 4.20 GHz
- •
GPU: NVIDIA RTX A6000 48GB
- •
Software: Python 3.11.8, PyTorch 2.2.2, Pennylane 0.35.1.

Based on the above computational infrastructure and setup, for example, our search framework can be reproduced in about 15 hours using 18 qubits.

### Use of llm s.

llms were used only to assist in polishing the language and improving readability. No part of the technical content, analysis, or experimental results was generated by llms.

## Peer Reviews

confidence: 4
contribution: 1
initial_rating: 2
presentation: 2
questions: 1.The experiments are limited to small-scale QNNs (≤20 qubits) and binary classification datasets. How would AdaInit perform on larger QNNs or multi-class/high-dimensional datasets?
2. AdaInit mainly ensures non-negligible initial gradient variance, but does it improve final training performance (accuracy, convergence speed, robustness/generalization)?Is higher initial gradient variance always correlated with better downstream training outcomes? Could the model still converge slowly or achieve suboptimal final accuracy?
3. How would AdaInit perform in noisy intermediate-scale quantum (NISQ) devices with measurement errors?
4.AdaInit relies on generative models (e.g., LLMs) to produce parameters. How sensitive is the method to different LLMs, model sizes, or prompt designs?Could errors or instability in the generated parameters negatively affect QNN training?
soundness: 3
strengths: 1.Utilizes generative models for parameter initialization, effectively ensuring non-negligible gradient variance in the early stage of training, thereby mitigating Barren Plateau (BP) issues.
2.Extensive experiments validate the framework’s effectiveness across different QNNs scales, demonstrating good scalability and adaptability.
3.The framework is adaptive, dynamically adjusting parameter generation based on dataset characteristics and gradient feedback, enhancing the intelligence of initialization.
4.Introduces a theoretically analyzable submartingale-based iterative process, providing formal guarantees for the convergence of generated parameters.
summary: The authors propose a foundational framework called AdaInit, which leverages generative models with a submartingale property to iteratively synthesize initial parameters for quantum neural networks (QNNs) that maintain non-negligible gradient variance, thereby effectively mitigating the Barren Plateau problem.
weaknesses: 1.Limited experimental scale: Currently validated only on small-scale QNNs with few qubits.Suggestion: Future work could explore larger-scale QNNs or real quantum hardware to evaluate scalability in high-dimensional quantum systems.
2.Focuses only on initialization: In practical applications, the key metrics are final training performance, including accuracy, convergence speed, and robustness/generalization. AdaInit only addresses the ability to start training by avoiding Barren Plateaus. Suggestion: To address this limitation, future work could extend AdaInit to not only ensure non-negligible initial gradient variance but also consider downstream training performance
3.Restricted dataset types: Only binary classification tasks were used, limiting generality.Suggestion: Test on multi-class or higher-dimensional datasets to verify robustness and applicability.
4.Dependency on generative model quality: The effectiveness of initialization relies on the output of the LLM or generative model.Suggestion: Introduce validation mechanisms or enhance the feedback loop to ensure generated parameters are effective.
5.High computational cost of iterative generation: Multiple iterations for parameter generation and gradient variance evaluation can be time-consuming.

---

confidence: 3
contribution: 2
initial_rating: 2
presentation: 3
questions: 1.Could you provide a more rigorous justification for why an LLM is a suitable tool for this task? What inductive biases or capabilities of modern LLMs make them superior to classical optimization algorithms like Bayesian Optimization or Evolutionary Strategies for exploring the QNN parameter space based on scalar feedback? 
2.How do you reconcile the apparent contradiction between the i.i.d. assumption used in your submartingale proofs and the history-dependent, iterative nature of the AdaInit algorithm? Could you clarify how the theoretical guarantees hold under the more realistic condition of temporal dependency in the parameter generation process?
soundness: 2
strengths: 1. Casts initialization as an iterative search with a simple progress statistic and connects it to sub martingale tools.
2. Empirical sweep spans qubits and layers across four datasets with ablations on prompts and llm hyperparams.
summary: The paper introduces AdaInit, designed to address the barren plateaus problem in QNN training. Unlike traditional methods that use static, preset distributions to initialize parameters, Adainit adopted an iterative strategy: it leverages a generative AI model to dynamically generate new initial parameters that are more likely to escape the BP problem. The paper theoretically models this iterative process as a sub martingale and proves its convergence. Experiments show that this method outperforms existing methods in maintaining high gradient variance, especially as the QNN scales.
weaknesses: 1.The core mechanism's assumptions are too strong. The paper's core argument rests on the assumption that large language models can intelligently explore the high-dimensional parameter space of QNNs, thereby finding regions of high gradient variance. This is an extremely strong assumption. The framework provides only a scalar feedback signal (gradient variance) from an extremely complex space. However, to date, there is no theoretical or empirical evidence demonstrating how LLMs can perform such high-dimensional, geometry-aware optimization.
2. Inconsistency between the theoretical framework and the proposed method: While the application of submartingale theory is ingenious, its underlying assumptions appear to contradict the very nature of the AdaInit framework. The generation of the new parameter θ(t) explicitly depends on historical data (θ(t-1) and S(t-1)), violating the independence assumption crucial to many theoretical assertions.
3. Critical omission of experimental noise: This paper aims to address the back propagation problem, a particularly acute problem in the NISQ era. However, the experiments in this paper are conducted in an idealized, noise-free simulation environment. This is a critical omission, as noise can create its own "barren plateaus" and fundamentally alter the gradient distribution. Methods that work in a noise-free environment are not guaranteed to work on real quantum hardware. The lack of experiments under realistic noise models severely limits the practical significance and impact of our results.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: The questions are included in the Weakness.
soundness: 2
strengths: The paper introduces a novel intersection of LLMs and quantum optimization, supported by rigorous theoretical proofs, but I have not checked them. The submartingale formulation is mathematically grounded, and experiments are clearly organized.
summary: This paper proposes AdaInit, an AI-driven framework that mitigates barren plateaus in quantum neural networks. The method iteratively generates effective initialization parameters using large language models (LLMs) guided by a submartingale-based process to ensure non-vanishing gradient variance. The authors provide theoretical guarantees for convergence and boundedness, and conduct experiments on various benchmark datasets, including Iris, Wine, Titanic, and MNIST.
weaknesses: 1. The use of a general-purpose LLM for updating parameter probabilities appears unjustified. It is unclear why an LLM not trained on quantum circuit data can reliably identify parameters that avoid barren plateaus.
2. The authors claimed the time complexity for QNN training is O(T_{tr} \cdot |\theta_0|), while overlooking the cost of calculating gradients in each step. In particular, evaluating gradients accurately may require exponentially many measurements, making the method potentially infeasible at larger scales.
3. The proposed algorithm also relies on computing the gradient variance $Var(\partial E^{(t)})$ at each iteration to guide parameter updates. Similarly, this calculation again demands exponentially many measurements under barren plateau conditions, raising concerns about its scalability and practicality on current NISQ hardware.
4. The manuscript includes numerous lemmas and theorems, some of which are peripheral to the main idea. This dense theoretical presentation obscures the core contributions. The paper would benefit from presenting only key theoretical results in the main text and moving technical proofs and auxiliary lemmas to the appendix for clarity.
5. The experimental setup is somewhat limited for studying barren plateaus, as the authors only explore configurations with either many qubits and shallow depth or few qubits and deep depth. Barren plateaus are theoretically less likely to occur in these regimes. To convincingly demonstrate the framework’s effectiveness, the authors should evaluate it on larger quantum circuits with both high qubit counts and significant depth, where barren plateaus are more probable and challenging to mitigate.

---

confidence: 4
contribution: 1
initial_rating: 2
presentation: 1
questions: - Line 118: Where is this definition of QNNs coming from? What kind of neural network layers are you talking about (repeated applications of U?). Where is the reference?
- Why does your architecture use a classical fully connected NN in the end? This is not standard to the best of my knowledge (otherwise provide reference). In any case, this should be stated in the main part of the paper.
- In light of the choice of adding a FCNN at the end of the quantum circuit, I would be curious to see how the performance of the models is if only the FCNN is used for prediction with the classical data only as input. As I said, I am questioning the whole setup, but even having QNN + classical FCNN to predict the simple datasets used seems already an overkill before employing the LLMs.
soundness: 1
strengths: - The authors test with a variety of different LLMs to evaluate differences in specific model used
- The authors put a lot of effort into providing the figures, which are as a result very intuitive to read
summary: The authors propose a framework based on LLMs to mitigate the BP phenomenon in QNNs. The authors provide a theoretical basis and first results to showcase their approach.

The paper has significant structural issues, in particular lacks a reading flow. The theorems and lemmas are stated as a long list without transitions and explanations, making the paper hard to follow. There is a substantial amount of literature on BPs missing and, considering that it is the main objective of the paper, needs to be added. Further, the related works are given as a list of papers, again, making it hard for a reader to actually grasp what the state-of-the-art is. 

The experimental setup is not clearly described, and the experiments are not extensive enough to conclude about the performance of the method. In particular, starting initialization only at five distinct points in time does not result in any sort of statistical significance. It is for this reason that I do not think that the manuscript is in a publishable format.

Further, I do not think that this paper provides any contribution to the broader research community. Using LLMs to aid in optimizing a QNN, whose task the LLMs could solve by themselves without any problem, seems to be a fundamental overkill. There is a significant waste of computational resources while prompting state-of-the-art (!) LLMs to optimize a QNN to solve iris, wine, Titanic and MNIST. These datasets are further not reasonable datasets for validating the approach, since even the simplest regression can solve them up to high accuracy. I am more than aware that it is not possible yet to really solve classically hard ML problems with QML, but at least moving beyond trivially solvable datasets would be appropriate.

I would urge the authors to rethink the objective of this work or provide a convincing reason why employing LLMs would be a valuable approach to training QNNs.
weaknesses: - "such as BPs, referring to a kind of..". This is really not a scientific description of a phenomenon, and in particular it is also not accurate towards the severity of the issues.
 - [Comment] please use \citep when citing within a sentence, this makes the paper more readable.
 - There is a substantial amount of literature on BPs missing. Given that the whole works aims at mitigating them, a clearer picture of state-of-the-art is necessary in the related work section. I would also expect more explanations of the approaches that are listed, s.t., the interested reader actually gets an overview of state-of-the-art, rather than just a collection of papers.
 - Line 110: "A quantum algorithm works by applying a sequence of unitary operators to an initial state". This is a description of a quantum circuit, not an algorithm. This is way too simplistic - it misses measurement, classical post-processing etc., which are crucial in quantum algorithms.
- Line 109: A state does not collapse into $\ket{i}$ for all $i$ after measurement, but rather into one of them. 
- Line 112: VQCs do not play a core role in QNNs - QNNs are a subcategory of VQCs. If your definition is different, please elaborate.
- From Definition 1. Please provide some connection between the theorems and some elaboration on them. Making the paper a pure list of theorems makes it unreadable. The same holds for Definition 2 onwards. I would also suggest providing at least proof sketches in the main part of the paper.
- Line 306: ansatz-induced BPs are not a thing (ansatz-induced could also mean entanglement induced, etc.). The referenced paper talks about the connection of expressivity (or more concretely approximate 2-designs) to BPs.
- Missing quantitative results. How much better are the approaches in percent, how significant are the results?
- The authors state that they repeat the results five times to get reliable results, however, this is not backed by statistical evidence. To get anything close to reliable, one would (1) need to incorporate significance tests and (2) run a much larger number of random initializations.
- The authors provide a trade-off analysis in the Appendix, to compare the computational costs and performance benefits. I am missing the discussion on usage of LLMs here.