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
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 L(θ) for an IQP generative model. The insets show the corresponding computational-basis distributions at representative initialization points: the model distribution pθ(z) is shown in gray, and the target distribution pdata(z) in green. Identity initialization ( θ∗=0 ) yields a highly biased distribution, pθ∗(0)=1 , which corresponds to a local maximum of the MMD. An unbiased data-agnostic initialization ( θj∗=π/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 pdata(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 σ∈Θ(n) . Curves correspond to different numbers of qubits n ranging from n=6 to n=16 . For the first three schemes, parameters are initialized as θj∼θj∗+Unif[−2πs,2πs] 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 L(θ) 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/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∈Θ(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 NE(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∪NE(A) . In this 5 qubits example, A={2,3} and NE(A)={1,4} .
Figure 5: Representation of a 1 -regular and a 2 -regular graph with different observables such that the maximum and minimum dA 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 σ∈O(1) . Curves correspond to different numbers of qubits n ranging from n=6 to n=16 . For the first three schemes, parameters are initialized as θj∼θj∗+Unif[−2πs,2πs] 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 .
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.
Gennaro De Luca, Vinayak Sharma, Aviral Shrivastava
We study a practical question in generative quantum machine learning: given a classical dataset, can we determine, before training, whether it is well suited to a quantum generative model? We focus on a class of quantum circuits known as instantaneous quantum polynomial-time (IQP) circuits, whose output distributions are widely believed to be difficult to sample from using classical methods. These circuits are used to build our quantum generative models. We introduce a Correlation-Complexity Map, a simple diagnostic built from two quantities computed from data samples. The first measures how closely the dataset's spectral correlation patterns resemble those naturally produced by IQP circuits, while the second quantifies how much of the dataset's structural correlation cannot be captured by simple pairwise models. In other words, we can estimate beforehand how well a dataset can be approximated by our model family and also how complex its correlations are, indicating possible failures of classical models. Applying this framework, we identify turbulence data as a promising target for quantum generative modeling. Guided by this analysis, we use a latent-parameter adaptation scheme that reuses a compact IQP circuit over a temporal sequence by learning and interpolating a low-dimensional latent trajectory, and observe competitive performance against classical baselines in a low-data, low-parameter regime. These results suggest that dataset-level diagnostics can help prioritize problems where quantum generative models are most likely to be useful, with improvements in data and parameter efficiency.
Chen-Yu Liu, Leonardo Placidi, Eric Brunner +1
Quantinuum, Partnership House, London, UK · Quantinuum, Otemachi Financial City Grand Cube, Tokyo, Japan
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), 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 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.