IQP Born Machines under Data-dependent and Agnostic Initialization Strategies

Sacha Lerch; Joseph Bowles; Ricard Puig; Erik Armengol; Zoë Holmes; Supanut Thanasilp

quant-phMar 15, 2026

IQP Born Machines under Data-dependent and Agnostic Initialization Strategies

Authors: Sacha Lerch, Joseph Bowles, Ricard Puig, Erik Armengol, Zoë Holmes, Supanut Thanasilp

Organizations: Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland · Centre for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland · Xanadu, Toronto, ON, M5G 2C8, Canada · ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain · Eurecat, Centre Tecnològic de Catalunya, Multimedia Technologies, 08005 Barcelona, Spain · Chula Intelligent and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand, 10330

Abstract

Quantum circuit Born machines based on instantaneous quantum polynomial-time (IQP) circuits are natural candidates for quantum generative modeling, both because of their probabilistic structure and because IQP sampling is provably classically hard in certain regimes. Recent proposals focus on training IQP-QCBMs using Maximum Mean Discrepancy (MMD) losses built from low-body Pauli- $Z$ correlators, but the effect of initialization on the resulting optimization landscape remains poorly understood. In this work, we address this by first proving that the MMD loss landscape suffers from barren plateaus for random full-angle-range initializations of IQP circuits. We then establish lower bounds on the loss variance for identity and an unbiased data-agnostic initialization. We then additionally consider a data-dependent initialization that is better aligned with the target distribution and, under suitable assumptions, yields provable gradients and generally converges quicker to a good minimum (as indicated by our training of circuits with 150 qubits on genomic data). Finally, as a by-product, the developed variance lower bound framework is applicable to a general class of non-linear losses, offering a broader toolset for analyzing warm-starts in quantum machine learning.

Figures & tables

Figure 1: Illustration of our main results. Schematic MMD loss landscape

\mathcal{L}(\boldsymbol{\theta})

for an IQP generative model. The insets show the corresponding computational-basis distributions at representative initialization points: the model distribution

p_{\boldsymbol{\theta}}(\boldsymbol{z})

is shown in gray, and the target distribution

p_{\rm data}(\boldsymbol{z})

in green. Identity initialization (

\boldsymbol{\theta}^{*}=\boldsymbol{0}

) yields a highly biased distribution,

p_{\boldsymbol{\theta}^{*}}(\boldsymbol{0})=1

, which corresponds to a local maximum of the MMD. An unbiased data-agnostic initialization (

\theta_{j}^{*}=\pi/4

) yields a uniform distribution over bit strings and usually corresponds to a region with gradient. A data-dependent initialization yields an initial distribution that is better aligned with the target. The green point indicates schematically the minimum corresponding to

p_{\rm data}(\boldsymbol{z})

Figure 2: Variance of the low-body MMD estimator versus initialization scale. Top row: Variance of the MMD estimator as a function of the initialization scale for four initialization schemes (identity, unbiased, data-dependent, and covariance) using a kernel with bandwidth that scales as

\sigma\in\Theta(\sqrt{n})

. Curves correspond to different numbers of qubits

n

ranging from

n=6

n=16

. For the first three schemes, parameters are initialized as

\theta_{j}\sim\theta_{j}^{*}+{\rm Unif}[-\frac{\pi}{2}s,\frac{\pi}{2}s]

where

s

is the initialization scale. The covariance initialization (right column) follows Eq. ( 33 ), where two-qubit gate parameters are correlated and perturbations are rescaled according to the target distribution covariances. The target distribution is given by a genomic dataset. Bottom row: Maximum variance over initialization scales (blue) and the initialization scale achieving this maximum (red dashed) as functions of the number of qubits

n

Figure 3: Training of the low-body MMD estimator for different initializations. Training loss $\mathcal{L}(\boldsymbol{\theta})$ as a function of gradient-descent iterations for a $n=150$ qubit model trained on the genomic dataset. Each curve corresponds to one of the four initialization strategies: identity (red), unbiased (blue), marginal matching (green), and the covariance-based data-dependent initialization (light green). The three panels correspond to different initialization scales $s$ (as defined in Fig. 2 ): $s=1/m$ , $s=1/\sqrt{m}$ , and $s=1$ , where $m$ denotes the total number of parameters. Each model is trained five times, with most initializations leading to similar overlapping loss curves except for the covariance-based initialization for $s\in\Theta(1)$ . For the linear and sqrt scalings, identity initialization shows an initial rapid decrease of the loss but quickly reaches a plateau at substantially larger loss values, leading to poor convergence compared to the other strategies. The unbiased initialization remains trainable for these scalings but converges more slowly than the data-dependent approaches. In contrast, under the unit scaling (corresponding to a full angle random initialization) the marginal-matching and unbiased initializations fail to train, whereas the covariance-based initialization can still achieve successful training in some trials, highlighting the benefit of incorporating parameter correlations in the initialization.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Illustration of the effective Pauli light cone. The subset $A$ is shown in green, and its external neighborhood $N_{E}(A)$ in red. The pink gates represent the interactions (equivalent to the edges of graph $E$ ), namely the $2$ -qubits gates in the IQP circuit. Gray qubits lie outside $A\cup N_{E}(A)$ . In this $5$ qubits example, $A=\{2,3\}$ and $N_{E}(A)=\{1,4\}$ .

Figure 5: Representation of a $1$ -regular and a $2$ -regular graph with different observables such that the maximum and minimum $d_{A}$ are obtained.

Figure 6: Empirical evaluation of Assumption 1 for the genomic dataset with

n=20

qubits.

Figure 7: Variance of the global MMD estimator versus initialization scale. Top row: Variance of the MMD estimator as a function of the initialization scale for four initialization schemes (identity, unbiased, data-dependent, and covariance) using a kernel with bandwidth that scales as

\sigma\in\mathcal{O}(1)

. Curves correspond to different numbers of qubits

n

ranging from

n=6

n=16

. For the first three schemes, parameters are initialized as

\theta_{j}\sim\theta_{j}^{*}+{\rm Unif}[-\frac{\pi}{2}s,\frac{\pi}{2}s]

where

s

is the initialization scale. The covariance initialization (right column) follows Eq. ( 262 ), where two-qubit gate parameters are correlated and perturbations are rescaled according to the target distribution covariances. The target distribution is given by a genomic dataset. Bottom row: Maximum variance over initialization scales (blue) and the initialization scale achieving this maximum (red dashed) as functions of the number of qubits

n

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Trainability of IQP Quantum Circuit Born Machines Under Gaussian Initialization

Gennaro De Luca, Vinayak Sharma, Aviral Shrivastava

SCAI, Arizona State University

Quantum Generative Modeling Quantum Machine Learning Barren Plateaus Variational Quantum Circuits Quantum Circuit Trainability

Open paper AskPDF arXiv

Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.

Abstract

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