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arXiv:2603.06440

Toward Generative Quantum Utility via Correlation-Complexity Map

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

cs.LGquant-phSubmitted Mar 6, 2026• Announced Mar 9, 2026• 0 citations
v1 · Mar 6, 2026v2 · May 5, 2026
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Abstract

We propose a Correlation-Complexity Map as a practical diagnostic tool for determining when real-world data distributions are structurally aligned with IQP-type quantum generative models. Characterized by two complementary indicators: (i) a Quantum Correlation-Likeness Indicator (QCLI), computed from the dataset's correlation-order (Walsh-Hadamard/Fourier) power spectrum aggregated by interaction order and quantified via Jensen-Shannon divergence from an i.i.d. binomial reference; and (ii) a Classical Correlation-Complexity Indicator (CCI), defined as the fraction of total correlation not captured by the optimal Chow-Liu tree approximation, normalized by total correlation. We provide theoretical support by relating QCLI to a support-mismatch mechanism, for fixed-architecture IQP families trained with an MMD objective, higher QCLI implies a smaller irreducible approximation floor. Using the map, we identify the classical turbulence data as both IQP-compatible and classically complex (high QCLI/high CCI). Guided by this placement, we use an invertible float-to-bitstring representation and a latent-parameter adaptation scheme that reuses a compact IQP circuit over a temporal sequence by learning and interpolating a low-dimensional latent trajectory. In comparative evaluations against classical models such as Restricted Boltzmann Machine (RBM) and Deep Convolutional Generative Adversarial Networks (DCGAN), the IQP approach achieves competitive distributional alignment while using substantially fewer training snapshots and a small latent block, supporting the use of QCLI/CCI as practical indicators for locating IQP-aligned domains and advancing generative quantum utility.

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33 pages, 8 figures

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We propose a Correlation-Complexity Map as a practical diagnostic tool for determining when real-world data distributions are structurally aligned with IQP-type quantum generative models. Characterized by two complementary indicators: (i) a Quantum Correlation-Likeness Indicator (QCLI), computed from the dataset's correlation-order (Walsh-Hadamard/Fourier) power spectrum aggregated by interaction order and quantified via Jensen-Shannon divergence from an i.i.d. binomial reference; and (ii) a Classical Correlation-Complexity Indicator (CCI), defined as the fraction of total correlation not captured by the optimal Chow-Liu tree approximation, normalized by total correlation. We provide theoretical support by relating QCLI to a support-mismatch mechanism, for fixed-architecture IQP families trained with an MMD objective, higher QCLI implies a smaller irreducible approximation floor. Using the map, we identify the classical turbulence data as both IQP-compatible and classically complex (high QCLI/high CCI). Guided by this placement, we use an invertible float-to-bitstring representation and a latent-parameter adaptation scheme that reuses a compact IQP circuit over a temporal sequence by learning and interpolating a low-dimensional latent trajectory. In comparative evaluations against classical models such as Restricted Boltzmann Machine (RBM) and Deep Convolutional Generative Adversarial Networks (DCGAN), the IQP approach achieves competitive distributional alignment while using substantially fewer training snapshots and a small latent block, supporting the use of QCLI/CCI as practical indicators for locating IQP-aligned domains and advancing generative quantum utility.
Quantum generative modelling casts sampling as a generative task: a parametrised quantum circuit is trained such that sampling reproduces a target probability distribution. Instantaneous Quantum Polynomial-time (IQP) circuits combine structural simplicity with complexity-theoretic evidence for quantum advantage. Yet their practical value depends not only on expressivity, but on how efficiently they consume genuinely quantum resources. We study this question through the lens of magic, or non-stabiliserness, as a resource for quantum generative modelling. We show that established fidelity- and geodesic-based notions of computational progress in a projective Hilbert space are ill-suited to generative models, since operational performance is determined by output probability distributions rather than quantum states themselves. We evaluate magic-consumption directly on the probability simplex, using changes in Jensen-Shannon divergence to quantify progress. Applying this framework to trained random γ-sparse IQP circuits shows signatures of efficient magic use, with the dominant contribution arising from two-qubit gates. As IQP circuits produce remarkably low intermediate magic relative to phase-randomised states with the same sampling distributions, this renders IQP-based quantum generative models as promising candidates for resource-efficient demonstrations of quantum advantage on early fault-tolerant architectures.
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Quantum generative modeling (QGM) relies on preparing quantum states and generating samples from these states as hidden---or known---probability distributions. As distributions from some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents an excellent test bed for quantum supremacy experiments. Furthermore, generative tasks are increasingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for demonstrating a practical quantum advantage. However, this requires that quantum circuits are trained to represent industrially relevant distributions, and the corresponding training stage has an extensive training cost for current quantum hardware in practice. In this work, we propose protocols for classical training of QGMs based on circuits of the specific type that admit an efficient gradient computation, while remaining hard to sample. In particular, we consider instantaneous quantum polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the time complexity, sparsity, and anticoncentration properties, we develop a classically tractable way of simulating their output probability distributions, allowing classical training to a target probability distribution. The corresponding quantum sampling from IQPs can be performed efficiently, unlike when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits using probability distributions for up to 30 qubits on a regular desktop computer. When applied to industrially relevant distributions this combination of classical training with quantum sampling represents an avenue for reaching advantage in the noisy intermediate-scale quantum (NISQ) era.
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.
Quantum generative learning based on instantaneous quantum polynomial-time (IQP) circuits can benefit from efficient classical training strategies. A recent latent adaptation framework for IQP-based generative modeling transfers shared circuit parameters across instances of the same task with different hyperparameters while adapting only a low-dimensional latent variable for each new instance. However, existing approaches initialize this latent variable randomly, which can limit optimization efficiency and performance. In this work, we introduce a physics-informed latent initialization scheme for IQP generative learning to improve upon existing random initialization schemes. Motivated by the platonic representation hypothesis, we use latent representations extracted from a classical physics-informed neural network (PINN) surrogate to initialize the latent variables of the quantum model for the solution of the Burgers' equation. The initialized IQP model is then adapted on a higher-resolution solution domain. We find that this structured initialization consistently outperforms random latent initialization, yielding improved adaptation behavior and stronger generative accuracy across multiple viscosity settings. These results show that classical surrogate representations can provide useful inductive bias for quantum generative models and offer a practical route to improved initialization in IQP-based learning.
We propose an approach to generative quantum machine learning that overcomes the fundamental scaling issues of variational quantum circuits. The core idea is to use a class of generative models based on instantaneous quantum polynomial circuits, which we show can be trained efficiently on classical hardware. Although training is classically efficient, sampling from these circuits is widely believed to be classically hard, and so computational advantages are possible when sampling from the trained model on quantum hardware. By combining our approach with a data-dependent parameter initialisation strategy, we do not encounter issues of barren plateaus and successfully circumvent the poor scaling of gradient estimation that plagues traditional approaches to quantum circuit optimisation. We investigate and evaluate our approach on a number of real and synthetic datasets, training models with up to one thousand qubits and hundreds of thousands of parameters. We find that the quantum models can successfully learn from high dimensional data, and perform surprisingly well compared to simple energy-based classical generative models trained with a similar amount of hyperparameter optimisation. Overall, our work demonstrates that a path to scalable quantum generative machine learning exists and can be investigated today at large scales.