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.
Figures & tables
Figure 1: Overview of the proposed correlation indicators and the resulting correlation–complexity map. (a) Quantum Correlation–Likeness Indicator (QCLI). Given a dataset of n -bit samples, we compute its correlation-order spectrum by aggregating Walsh–Hadamard (Fourier–Walsh) power over subset sizes k=∣s∣ . QCLI, IQCLI , is defined as the Jensen–Shannon divergence between this empirical order spectrum and the binomial reference spectrum induced by the uniform (classically random) distribution. (b) Classical Correlation–Complexity Indicator (CCI). For the same dataset, we compute the total correlation (multi-information) and the dependence captured by the optimal Chow–Liu tree, obtained by maximizing ∑(i,j)∈TI(Xi;Xj) over spanning trees T . CCI, ICCI , is the fraction of total correlation not explained by the optimal tree-structured model. (c) Correlation–Complexity Map. Datasets are shown in the (IQCLI,ICCI) plane, separating regimes that are locally structured and classically expressible (low ICCI ), quantum-like but classically easy, high-order but non-IQP-aligned, and the high- IQCLI /high- ICCI regime that is most compatible with IQP-type generative inductive biases. The turbulence dataset is highlighted as residing in this IQP-compatible quadrant, motivating the IQP-based generative modeling experiments. (d) Latent adaptation for temporal turbulence synthesis. Starting from a trained IQP generator, we fix a shared core parameter block and assign (adapt) a low-dimensional latent block to represent different time snapshots. Interpolating or extrapolating the learned latent trajectory enables generation of unseen turbulence snapshots while keeping the circuit architecture and core parameters fixed.
Figure 2: Correlation-order spectra and IQCLI on D-Wave 100Q datasets [ 41 ] . Left: the empirical correlation-order spectrum mk (spectral mass aggregated over Walsh–Hadamard parities of order ∣s∣=k ) for three D-Wave 100Q datasets collected at anneal times τ∈{1,10,100}μs , with the corresponding IQCLI shown in the titles. Right: the absolute deviation ∣mk−bk∣ from the i.i.d. ideal binomial baseline spectrum bk . As τ increases, IQCLI rises and the spectra become visibly more structured, exhibiting sharper peaks and pronounced cancellations across correlation orders, consistent with stronger constructive/destructive interference patterns.
Figure 3: Empirical probe of the QCLI–MMD support-mismatch mechanism. Scatter plots show the MMD loss achieved when fitting a fixed-architecture IQP learner to a collection of IQP-generated datasets spanning a wide range of IQCLI values, under a fixed training budget. Each panel corresponds to a different learner expressivity setting (gate count Gn∈{50,100,150} , with restricted locality), thereby enforcing a controlled architecture–data mismatch. Dashed curves indicate the estimated upper and lower envelopes of the observed MMD as a function of IQCLI . Details of the envelope estimation are provided in Appendix D .
Figure 4: IQCLI versus ICCI for 2-local IQP circuits with 8Q, 12Q, and 16Q, each using 150 random gates. Circuit parameters are optimized solely to maximize IQCLI , yet higher IQCLI consistently coincides with elevated ICCI across all system sizes, indicating an emergent coupling between IQP representational alignment and global multivariate classical correlations.
Figure 5: Correlation–Complexity Map. Each dataset is positioned by (IQCLI,ICCI) , measuring (x-axis) parity-structured deviation of the correlation-order spectrum from an i.i.d. binomial baseline (QCLI) and (y-axis) the fraction of total dependence not captured by the optimal Chow–Liu tree ( ICCI ). The shaded region shows a protocol-specific empirical IQP envelope obtained by optimizing 16 -qubit IQP circuits for high IQCLI and ICCI . Datasets in the upper-right quadrant exhibit both interference-like spectral structure and beyond-pairwise dependence, forming the regime most structurally compatible with IQP-type generators. Point colors indicate data provenance: orange denotes datasets produced by classical processes (e.g., simulations or classical benchmarks), whereas teal denotes datasets generated by quantum processes or quantum hardware.
Dataset (size)
IQP setup
Baselines
Key outcome
BinaryBlobs (16 bits)
Local IQP gates (up to 6-body), 14892 params
EBM (1700 params), RBM, Bitflip
EBM achieves better likelihood than IQP (log-likelihood: EBM −5.34 vs. IQP −6.35 ); Bitflip performs poorly; MMD reveals sensitivity to mode imbalance.
D-Wave annealer samples (484 qubits, 100μs )
All 1- and 2-qubit gates on 484 qubits, >105 params
RBM, EBM, Bitflip
No model fits perfectly, but IQP performs best among the compared models; RBM/EBM can score worse than random on test MMD; EBM fails in this setting.
Binarized MNIST (784 bits)
All-to-all 2-qubit gates, 307,720 params
Bitflip only
IQP captures substantial structure; covariance matches the overall pattern, albeit more weakly. Test MMD is comparable to a noisy-data reference (reported around noise level p≈0.3 ); Bitflip fails to learn comparable structure.
Table 1: Summary of representative IQP generative modeling results reported in Ref. [ 38 ] for datasets (or similar classes) that also appear in our correlation–complexity map. We use this prior evidence to contextualize why we do not retrain IQP models for every dataset in Fig. 5 , and to motivate focusing new experiments on the high- IQCLI /high- ICCI turbulence regime.
Algorithm 1 Latent-parameter adaptation for turbulence synthesis
Figure 6: Latent-parameter adaptation and interpolation for IQP-based generative synthesis of turbulence snapshots. Step 1: Fit a single IQP generator to the reference snapshot at t=1 by optimizing the shared core parameters θcore while holding the latent block θlat(1) fixed at its random initialization. Step 2: Freeze θcore(1) and adapt only the low-dimensional latent parameters θlat to match the next snapshot (t=2) , yielding θlat(2) . Step 3: Repeat the latent-only adaptation sequentially for t=3,4,…,T to obtain a latent trajectory {θlat(t)}t=1T while reusing the same fixed core circuit. Step 4: Generate snapshots at unseen times τ by assigning latent parameters via interpolation/extrapolation θ~lat(τ) (e.g., piecewise-linear interpolation between neighboring θlat(t)) , and sampling from q(θcore(1),θ~lat(τ)) followed by decoding to the physical turbulence field. The time indices (e.g.,t=2,3,…,10) and turbulence snapshots shown are for illustration only; our experiments use a different subset of observed snapshots, with the same adaptation and interpolation procedure applied unchanged.
Figure 7: Learned latent-parameter trajectory across snapshot adaptation steps. Each curve shows one latent coordinate (θlat(t))j as a function of the snapshot index t∈{1,100,200,300,400,500,600,700,800,900,1000} , obtained by sequential latent-only adaptation while keeping the shared core parameters θcore(1) fixed. All dlat=50 latent coordinates are plotted. The gradually varying evolution across t supports viewing {θlat(t)} as a low-dimensional trajectory and this trajectory motivates interpolation/extrapolation θ~lat(τ) to synthesize turbulence snapshots at unseen times.
Table 2: Qualitative samples (center) with corresponding PDF–JS and feature-space MMD 2 scores (right). Lower scores indicate better alignment with the real turbulence distribution. Models are shown with their parameter counts in parentheses.
Figure 8: DCGAN data-scaling versus IQP baselines on turbulence. Left: PDF–JS divergence; right: Conv2D feature-space MMD (lower is better), both evaluated by comparing 200 real snapshots to 200 generated samples. Solid lines show DCGAN performance as a function of the number of training snapshots Ntrain∈{10,30,50,70,90,100} for three model sizes (51k, 0.2M, 1.2M parameters); error bars indicate variability across independent runs. Horizontal dashed lines denote the IQP reference performance under the latent-adaptation setting trained on 11 snapshots (the same setting used in Tables 2 ).
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.
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.
Sacha Lerch, Joseph Bowles, Ricard Puig +3
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 +3
Many applications in quantum simulation, quantum chemistry, and quantum machine learning require not a single quantum state but an ensemble of states characterizing the heterogeneity of a target system. Preparing such ensembles state-by-state is prohibitive in both variational and fault-tolerant settings, thereby motivating a generative modeling approach. We introduce latent-conditioned parameterized quantum circuits (LPQCs), a hybrid quantum-classical framework in which classical neural networks map a latent variable sampled from a prior distribution to the parameters of a parameterized quantum circuit. We prove that LPQCs are universal approximators for probability measures over density operators in the 1-Wasserstein distance, extending classical universal approximation theorems to the quantum-distribution setting. We additionally introduce a multimodal latent prior and a mixture-of-experts circuit architecture, and show empirically that the latent-conditioned parameterization alleviates the barren plateau problem during optimization, a behavior for which we provide rigorous partial guarantees. Numerical experiments validate the framework on a synthetic multi-cluster ensemble of mixed quantum states and on a QM9-derived ensemble of 3-D molecular structures. In these tasks, LPQC outperforms recent quantum generative baselines and matches the generation quality of a classical neural-network baseline, while requiring an output dimension that grows only linearly with the number of qubits rather than exponentially. By leveraging classical expressivity in the latent space, LPQCs offer a tractable route to quantum generative modeling.
Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo +1
Quantum Laboratory, Fujitsu Research, Fujitsu Limited, Kawasaki, Kanagawa 211-8588, Japan