Toward Generative Quantum Utility via Correlation-Complexity Map

Chen-Yu Liu; Leonardo Placidi; Eric Brunner; Enrico Rinaldi

cs.LGMar 6, 2026

Toward Generative Quantum Utility via Correlation-Complexity Map

Authors: Chen-Yu Liu, Leonardo Placidi, Eric Brunner, Enrico Rinaldi

Organizations: Quantinuum, Partnership House, London, UK · Quantinuum, Otemachi Financial City Grand Cube, Tokyo, Japan

Abstract

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, $I_{\text{QCLI}}$ , 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 $\sum_{(i,j)\in T}I(X_{i};X_{j})$ over spanning trees $T$ . CCI, $I_{\text{CCI}}$ , is the fraction of total correlation not explained by the optimal tree-structured model. (c) Correlation–Complexity Map. Datasets are shown in the $(I_{\text{QCLI}},I_{\text{CCI}})$ plane, separating regimes that are locally structured and classically expressible (low $I_{\text{CCI}}$ ), quantum-like but classically easy, high-order but non-IQP-aligned, and the high- $I_{\text{QCLI}}$ /high- $I_{\text{CCI}}$ 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 $I_{\text{QCLI}}$ on D-Wave 100Q datasets [ 41 ] . Left: the empirical correlation-order spectrum $m_{k}$ (spectral mass aggregated over Walsh–Hadamard parities of order $|s|=k$ ) for three D-Wave 100Q datasets collected at anneal times $\tau\in\{1,10,100\}\,\mu\text{s}$ , with the corresponding $I_{\text{QCLI}}$ shown in the titles. Right: the absolute deviation $|m_{k}-b_{k}|$ from the i.i.d. ideal binomial baseline spectrum $b_{k}$ . As $\tau$ increases, $I_{\text{QCLI}}$ 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 $I_{\text{QCLI}}$ values, under a fixed training budget. Each panel corresponds to a different learner expressivity setting (gate count $G_{n}\in\{50,100,150\}$ , 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 $I_{\text{QCLI}}$ . Details of the envelope estimation are provided in Appendix D .

Figure 4: $I_{\text{QCLI}}$ versus $I_{\text{CCI}}$ for 2-local IQP circuits with 8Q, 12Q, and 16Q, each using 150 random gates. Circuit parameters are optimized solely to maximize $I_{\text{QCLI}}$ , yet higher $I_{\text{QCLI}}$ consistently coincides with elevated $I_{\text{CCI}}$ 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 $(I_{\text{QCLI}},I_{\text{CCI}})$ , 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 ( $I_{\text{CCI}}$ ). The shaded region shows a protocol-specific empirical IQP envelope obtained by optimizing $16$ -qubit IQP circuits for high $I_{\text{QCLI}}$ and $I_{\text{CCI}}$ . 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, $100\,\mu\mathrm{s}$ )	All 1- and 2-qubit gates on 484 qubits, $>10^{5}$ 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 $p\approx 0.3$ ); 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-

I_{\mathrm{QCLI}}

/high-

I_{\mathrm{CCI}}

turbulence regime.

\theta_{\mathrm{core}}^{(1)}\leftarrow\arg\min_{\theta_{\mathrm{core}}}\mathcal{L}_{\mathrm{MMD}}\bigl(q_{(\theta_{\mathrm{core}},\theta_{\mathrm{lat}}^{(t_{1})})},\hat{p}_{t_{1}}\bigr),

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 $\theta_{\mathrm{core}}$ while holding the latent block $\theta_{\mathrm{lat}}^{(1)}$ fixed at its random initialization. Step 2: Freeze $\theta_{\mathrm{core}}^{(1)}$ and adapt only the low-dimensional latent parameters $\theta_{\mathrm{lat}}$ to match the next snapshot $(t=2)$ , yielding $\theta_{\mathrm{lat}}^{(2)}$ . Step 3: Repeat the latent-only adaptation sequentially for $t=3,4,\ldots,T$ to obtain a latent trajectory $\{\theta_{\mathrm{lat}}^{(t)}\}_{t=1}^{T}$ while reusing the same fixed core circuit. Step 4: Generate snapshots at unseen times $\tau$ by assigning latent parameters via interpolation/extrapolation $\tilde{\theta}_{\mathrm{lat}}(\tau)$ (e.g., piecewise-linear interpolation between neighboring $\theta_{\mathrm{lat}}^{(t)})$ , and sampling from $q_{(\theta_{\mathrm{core}}^{(1)},\tilde{\theta}_{\mathrm{lat}}(\tau))}$ followed by decoding to the physical turbulence field. The time indices $(e.g.,t=2,3,\ldots,10)$ 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 $\bigl(\theta_{\mathrm{lat}}^{(t)}\bigr)_{j}$ as a function of the snapshot index $t\in\{1,100,200,300,400,500,600,700,800,900,1000\}$ , obtained by sequential latent-only adaptation while keeping the shared core parameters $\theta_{\mathrm{core}}^{(1)}$ fixed. All $d_{\mathrm{lat}}=50$ latent coordinates are plotted. The gradually varying evolution across $t$ supports viewing $\{\theta_{\mathrm{lat}}^{(t)}\}$ as a low-dimensional trajectory and this trajectory motivates interpolation/extrapolation $\tilde{\theta}_{\mathrm{lat}}(\tau)$ to synthesize turbulence snapshots at unseen times.

Panel of Table 2 — 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 $N_{\text{train}}\in\{10,30,50,70,90,100\}$ 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 ).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Item	Setting
Target datasets	$10^{4}$ samples from $n=16$ IQP circuits
Data-generating circuits	Locality $\leq 4$ , $G_{n}^{\mathrm{data}}\in\{140,280,420,560,700,840,1050\}$
Target diversity	SPSA-optimized parameters spanning a range of $I_{\mathrm{QCLI}}$
Learner circuits	Restricted IQP, locality $\leq 2$ , $G_{n}\in\{50,100,150\}$
Training	Train-on-classical/deploy-on-quantum; Adam, $5000$ iterations, lr $10^{-4}$
Evaluation	Achieved MMD loss after training

Appendix

Table 3: Settings for the controlled QCLI–MMD support-mismatch probe.

Item	Setting
System sizes	$n\in\{8,12,16\}$ qubits
Circuit architecture	IQP circuits with $150$ randomly selected 2-local parameterized gates
Initialization and optimizer	Random initialization; SPSA optimization
Objective	Maximize $I_{\mathrm{QCLI}}$ only; no $I_{\mathrm{CCI}}$ term used
Sampling along optimization	$20000$ IQP circuits per system size
Evaluation	Compute $I_{\mathrm{QCLI}}$ and $I_{\mathrm{CCI}}$ for each sampled circuit

Appendix

Table 4: Settings for the high-

I_{\mathrm{QCLI}}

IQP output and

I_{\mathrm{CCI}}

experiment.

Item	Setting
Dataset	D2 turbulence dataset from Ref. [ 25 ]
Number of snapshots	$1000$
Time indices	$t\in\{1,2,\ldots,1000\}$
Basic data unit	Coordinate/value tuple $(x,y,z)\in\mathbb{R}^{3}$
Quantization	$N=6$ bits per coordinate/value component
Bitstring length	$3N=18$ bits per tuple

Appendix

Table 5: Turbulence dataset and representation used in the IQP generative modeling experiments.

Item	Setting
Training snapshots	$11$ anchor snapshots
Anchor time indices	$t\in\{1,100,200,300,400,500,600,700,800,900,1000\}$
Parameter partition	Shared core block $\theta_{\mathrm{core}}$ and latent block $\theta_{\mathrm{lat}}$
Latent dimension	$d_{\mathrm{lat}}=50$
First snapshot training	$\theta_{\mathrm{lat}}^{(1)}$ is randomly initialized and fixed; $\theta_{\mathrm{core}}$ is trained by minimizing MMD
First-stage training data	$N_{t}=5\times 10^{4}$ training datapoints

Appendix

Table 6: IQP latent-adaptation settings for turbulence generation.

Item	Setting
Training snapshots	Same $11$ training snapshots as the IQP model
Data representation	Same $18$ -bit coordinate/value tuple representation as the IQP model
Training domain	Bitstring domain
Latent dimension	$d_{\mathrm{lat}}=50$
Latent block	First $d_{\mathrm{lat}}=50$ entries of a fixed flattened RBM parameter ordering
Fixed parameters	Remaining RBM parameters are held fixed analogously to the IQP case

Appendix

Table 7: RBM baseline settings for the turbulence generative modeling experiments.

Item	Setting
Data representation	Quantized two-dimensional turbulence fields, not the bitstring representation
Data-matched regime	$11$ training snapshots, matching IQP/RBM
Data-rich regime	$100$ training snapshots
Model sizes	Approximately $51$ k, $0.2$ M, and $1.2$ M parameters
Latent dimension	$128$
Data-scaling study	$N_{\mathrm{train}}\in\{10,30,50,70,90,100\}$ snapshots

Appendix

Table 8: DCGAN baseline settings for the turbulence generative modeling experiments.

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Aug 31, 2026J/K move · Enter open · S save

"Train classical, deploy quantum" requires rethinking generalization

Snehal Raj, Natansh Mathur, Alejandro Perdomo-Ortiz

QC Ware Corp., Palo Alto, USA and Paris, France · LIP6, CNRS, Sorbonne Universit´e, Paris, France

Quantum Generative Modeling

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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 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.

Abstract

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Latent-Conditioned Parameterized Quantum Circuits as Universal Approximators for Distributions over Quantum States