Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.08476

Universality and kernel-adaptive training for classically trained, quantum-deployed generative models

Andrii Kurkin, Shen, Kevin, Susanne Pielawa, Hao Wang, Vedran Dunjko

Published Oct 9, 2025• 0 citations
arXivHTMLPDF

Abstract

The instantaneous quantum polynomial (IQP) quantum circuit Born machine (QCBM) has been proposed as a promising quantum generative model over bitstrings. Recent works have shown that the training of IQP-QCBM is classically tractable w.r.t. the so-called Gaussian kernel maximum mean discrepancy (MMD) loss function, while maintaining the potential of a quantum advantage for sampling itself. Nonetheless, the model has a number of aspects where improvements would be important for more general utility: (1) the basic model is known to be not universal - i.e. it is not capable of representing arbitrary distributions, and it was not known whether it is possible to achieve universality by adding hidden (ancillary) qubits; (2) a fixed Gaussian kernel used in the MMD loss can cause training issues, e.g., vanishing gradients. In this paper, we resolve the first question and make decisive strides on the second. We prove that for an $n$-qubit IQP generator, adding $n + 1$ hidden qubits makes the model universal. For the latter, we propose a kernel-adaptive training method, where the kernel is adversarially trained. We show that in the kernel-adaptive method, the convergence of the MMD value implies weak convergence in distribution of the generator. We also analytically analyze the limitations of the MMD-based training method. Finally, we verify the performance benefits on the dataset crafted to spotlight improvements by the suggested method. The results show that kernel-adaptive training outperforms a fixed Gaussian kernel in total variation distance, and the gap increases with the dataset dimensionality. These modifications and analyses shed light on the limits and potential of these new quantum generative methods, which could offer the first truly scalable insights in the comparative capacities of classical versus quantum models, even without access to scalable quantum computers.

DOI
10.48550/arxiv.2510.08476

Related papers

Characterizing Trainability of Instantaneous Quantum Polynomial Circuit Born Machines

Kevin Shen, Susanne Pielawa, Vedran Dunjko +1 more

Instantaneous quantum polynomial quantum circuit Born machines (IQP-QCBMs) have been proposed as quantum generative models with a classically tractable training objective based on the maximum mean discrepancy (MMD) and a potential quantum advantage motivated by sampling-complexity arguments, making them an exciting model worth deeper investigation. While recent works have further proven the universality of a (slightly generalized) model, the next immediate question pertains to its trainability, i.e., whether it suffers from the exponentially vanishing loss gradients, known as the barren plateau issue, preventing effective use, and how regimes of trainability overlap with regimes of possible quantum advantage. Here, we provide significant strides in these directions. To study the trainability at initialization, we analytically derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and provide general upper and lower bounds. With uniform initialization, we show that barren plateaus depend on the generator set and the spectrum of the chosen kernel. We identify regimes in which low-weight-biased kernels avoid exponential gradient suppression in structured topologies. Also, we prove that a small-variance Gaussian initialization ensures polynomial scaling for the gradient under mild conditions. As for the potential quantum advantage, we further argue, based on previous complexity-theoretic arguments, that sparse IQP families can output a probability distribution family that is classically intractable, and that this distribution remains trainable at initialization at least at lower-weight frequencies.

quant-pharXiv:2602.11042•Feb 12, 2026•0 citations

Trainability of IQP Quantum Circuit Born Machines Under Gaussian Initialization

Gennaro De Luca, Vinayak Sharma, Aviral Shrivastava

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.

quant-pharXiv:2606.10179•Jun 10, 2026•0 citations

Universality of Classically Trainable, Quantum-Deployed Boson-Sampling Generative Models

Andrii Kurkin, Ulysse Chabaud, Zoltán Kolarovszki +3 more

Recent work on the instantaneous quantum polynomial-time (IQP) quantum-circuit Born machine (QCBM) highlights a promising paradigm for generative modeling: train classically, deploy quantumly. In this setting, the training objective can be evaluated efficiently on a classical computer, while sampling from the resulting model may still be classically intractable. Furthermore, in the IQP-QCBM framework, extending the model family with ancillary qubits has been proven to yield universality. This paper asks whether similar results hold for linear-optical generative models. To this end, we introduce the Boson Sampling Born Machine (BSBM). Our analysis retraces analogous steps as were found for IQP-QCBMs with twists. Using recent results that enable classical approximation of broad classes of expectation values in linear optics, we show that BSBMs can be trained classically for wide families of loss functions. Next, we argue that "basic" BSBMs are not universal generative models, and that universality can be achieved by expanding the model while preserving efficient classical training and sampling hardness. In our approach, we introduce and analyze the role of constant-function postprocessing, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models. We showcase a family of BSBMs, characterized by a single hyperparameter, that allows for a monotonic increase in expressivity toward universality while retaining the capacity to represent ostensibly hard distributions. Furthermore, we discuss the possible modalities for the efficient classical training, in the sense of efficient estimation of gradients of the loss function.

quant-pharXiv:2603.11014•Mar 12, 2026•0 citations

IQP Born Machines under Data-dependent and Agnostic Initialization Strategies

Sacha Lerch, Joseph Bowles, Ricard Puig +3 more

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.

quant-pharXiv:2603.14576•Mar 17, 2026•1 citation

"Train classical, deploy quantum" requires rethinking generalization

Snehal Raj, Natansh Mathur, Alejandro Perdomo-Ortiz

Generative models have become central across science and industry, from image and text synthesis to the design of molecules and materials. Quantum generative models are considered one of the most promising applications for quantum computers, since a quantum circuit naturally produces samples from the distribution it encodes, and for suitable circuits that distribution is believed to be hard for any classical computer to reproduce. A leading strategy trains these models on a classical computer and reserves the quantum device for generating samples at deployment. This is possible when the training loss can be evaluated on a classical computer. A prime example is the maximum mean discrepancy (MMD$^2$), a moment-matching loss that compares the model and the data through their Pauli-$Z$ correlations. Research so far has asked whether such models can be trained and whether their sampling is hard; whether minimizing such an objective yields a model that \emph{generalizes}, rather than one that merely reproduces the training statistics, remains poorly understood. We benchmark thirteen quantum and classical generative models by direct sampling on two application-inspired datasets: first a cardinality-constrained dataset at up to $30$ qubits and second a dataset of genomic single-nucleotide variants, whose valid set is the observed data. Models that converge the loss to the same value differ widely in how much of the unseen valid set they cover. These results indicate that a converged moment-matching loss is not a reliable measure of generalization, and that a train-classical, deploy-quantum workflow has to measure generalization by sampling the trained model, a step that at the sizes of interest is believed to require the quantum device.

quant-pharXiv:2608.31117•Sep 1, 2026•0 citations

Differentiable learning of quantum circuit Born machines

Jin-Guo Liu, Lei Wang

Quantum circuit Born machines are generative models which represent the probability distribution of classical dataset as quantum pure states. Computational complexity considerations of the quantum sampling problem suggest that the quantum circuits exhibit stronger expressibility compared to classical neural networks. One can efficiently draw samples from the quantum circuits via projective measurements on qubits. However, similar to the leading implicit generative models in deep learning, such as the generative adversarial networks, the quantum circuits cannot provide the likelihood of the generated samples, which poses a challenge to the training. We devise an efficient gradient-based learning algorithm for the quantum circuit Born machine by minimizing the kerneled maximum mean discrepancy loss. We simulated generative modeling of the BARS-AND-STRIPES dataset and Gaussian mixture distributions using deep quantum circuits. Our experiments show the importance of circuit depth and the gradient-based optimization algorithm. The proposed learning algorithm is runnable on near-term quantum device and can exhibit quantum advantages for probabilistic generative modeling.

arXiv:1804.04168•Physical Review A•Dec 19, 2018•279 citations
The instantaneous quantum polynomial (IQP) quantum circuit Born machine (QCBM) has been proposed as a promising quantum generative model over bitstrings. Recent works have shown that the training of IQP-QCBM is classically tractable w.r.t. the so-called Gaussian kernel maximum mean discrepancy (MMD) loss function, while maintaining the potential of a quantum advantage for sampling itself. Nonetheless, the model has a number of aspects where improvements would be important for more general utility: (1) the basic model is known to be not universal - i.e. it is not capable of representing arbitrary distributions, and it was not known whether it is possible to achieve universality by adding hidden (ancillary) qubits; (2) a fixed Gaussian kernel used in the MMD loss can cause training issues, e.g., vanishing gradients. In this paper, we resolve the first question and make decisive strides on the second. We prove that for an nnn-qubit IQP generator, adding n+1n + 1n+1 hidden qubits makes the model universal. For the latter, we propose a kernel-adaptive training method, where the kernel is adversarially trained. We show that in the kernel-adaptive method, the convergence of the MMD value implies weak convergence in distribution of the generator. We also analytically analyze the limitations of the MMD-based training method. Finally, we verify the performance benefits on the dataset crafted to spotlight improvements by the suggested method. The results show that kernel-adaptive training outperforms a fixed Gaussian kernel in total variation distance, and the gap increases with the dataset dimensionality. These modifications and analyses shed light on the limits and potential of these new quantum generative methods, which could offer the first truly scalable insights in the comparative capacities of classical versus quantum models, even without access to scalable quantum computers.
Instantaneous quantum polynomial quantum circuit Born machines (IQP-QCBMs) have been proposed as quantum generative models with a classically tractable training objective based on the maximum mean discrepancy (MMD) and a potential quantum advantage motivated by sampling-complexity arguments, making them an exciting model worth deeper investigation. While recent works have further proven the universality of a (slightly generalized) model, the next immediate question pertains to its trainability, i.e., whether it suffers from the exponentially vanishing loss gradients, known as the barren plateau issue, preventing effective use, and how regimes of trainability overlap with regimes of possible quantum advantage. Here, we provide significant strides in these directions. To study the trainability at initialization, we analytically derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and provide general upper and lower bounds. With uniform initialization, we show that barren plateaus depend on the generator set and the spectrum of the chosen kernel. We identify regimes in which low-weight-biased kernels avoid exponential gradient suppression in structured topologies. Also, we prove that a small-variance Gaussian initialization ensures polynomial scaling for the gradient under mild conditions. As for the potential quantum advantage, we further argue, based on previous complexity-theoretic arguments, that sparse IQP families can output a probability distribution family that is classically intractable, and that this distribution remains trainable at initialization at least at lower-weight frequencies.
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.
Recent work on the instantaneous quantum polynomial-time (IQP) quantum-circuit Born machine (QCBM) highlights a promising paradigm for generative modeling: train classically, deploy quantumly. In this setting, the training objective can be evaluated efficiently on a classical computer, while sampling from the resulting model may still be classically intractable. Furthermore, in the IQP-QCBM framework, extending the model family with ancillary qubits has been proven to yield universality. This paper asks whether similar results hold for linear-optical generative models. To this end, we introduce the Boson Sampling Born Machine (BSBM). Our analysis retraces analogous steps as were found for IQP-QCBMs with twists. Using recent results that enable classical approximation of broad classes of expectation values in linear optics, we show that BSBMs can be trained classically for wide families of loss functions. Next, we argue that "basic" BSBMs are not universal generative models, and that universality can be achieved by expanding the model while preserving efficient classical training and sampling hardness. In our approach, we introduce and analyze the role of constant-function postprocessing, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models. We showcase a family of BSBMs, characterized by a single hyperparameter, that allows for a monotonic increase in expressivity toward universality while retaining the capacity to represent ostensibly hard distributions. Furthermore, we discuss the possible modalities for the efficient classical training, in the sense of efficient estimation of gradients of the loss function.
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-ZZZ 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.
Generative models have become central across science and industry, from image and text synthesis to the design of molecules and materials. Quantum generative models are considered one of the most promising applications for quantum computers, since a quantum circuit naturally produces samples from the distribution it encodes, and for suitable circuits that distribution is believed to be hard for any classical computer to reproduce. A leading strategy trains these models on a classical computer and reserves the quantum device for generating samples at deployment. This is possible when the training loss can be evaluated on a classical computer. A prime example is the maximum mean discrepancy (MMD2^22), a moment-matching loss that compares the model and the data through their Pauli-ZZZ correlations. Research so far has asked whether such models can be trained and whether their sampling is hard; whether minimizing such an objective yields a model that \emph{generalizes}, rather than one that merely reproduces the training statistics, remains poorly understood. We benchmark thirteen quantum and classical generative models by direct sampling on two application-inspired datasets: first a cardinality-constrained dataset at up to 303030 qubits and second a dataset of genomic single-nucleotide variants, whose valid set is the observed data. Models that converge the loss to the same value differ widely in how much of the unseen valid set they cover. These results indicate that a converged moment-matching loss is not a reliable measure of generalization, and that a train-classical, deploy-quantum workflow has to measure generalization by sampling the trained model, a step that at the sizes of interest is believed to require the quantum device.
Quantum circuit Born machines are generative models which represent the probability distribution of classical dataset as quantum pure states. Computational complexity considerations of the quantum sampling problem suggest that the quantum circuits exhibit stronger expressibility compared to classical neural networks. One can efficiently draw samples from the quantum circuits via projective measurements on qubits. However, similar to the leading implicit generative models in deep learning, such as the generative adversarial networks, the quantum circuits cannot provide the likelihood of the generated samples, which poses a challenge to the training. We devise an efficient gradient-based learning algorithm for the quantum circuit Born machine by minimizing the kerneled maximum mean discrepancy loss. We simulated generative modeling of the BARS-AND-STRIPES dataset and Gaussian mixture distributions using deep quantum circuits. Our experiments show the importance of circuit depth and the gradient-based optimization algorithm. The proposed learning algorithm is runnable on near-term quantum device and can exhibit quantum advantages for probabilistic generative modeling.