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arXiv:2511.05267 · Physical Review A

Shallow IQP circuits and graph generation

Oriol Balló-Gimbernat, Marcos Arroyo-Sánchez, Paula García-Molina, Adan Garriga, Fernando Vilariño

quant-phSubmitted Nov 7, 2025• Announced Nov 10, 2025• 0 citations• Revised Nov 15, 2025 (v2)
v1 · Nov 7, 2025v2 · Nov 15, 2025v3 · Mar 12, 2026
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Abstract

We introduce shallow instantaneous quantum polynomial-time (IQP) circuits as generative graph models, using an edge-qubit encoding to map graphs onto quantum states. Focusing on bipartite and Erdős-Rényi distributions, we study their expressivity and robustness through simulations and large-scale experiments. Noiseless simulations of $28$ qubits ($8$-node graphs) reveal that shallow IQP models can learn key structural features, such as the edge density and bipartite partitioning. On IBM's Aachen QPU, we scale experiments from $28$ to $153$ qubits ($8$-$18$ nodes) in order to characterize performance on real quantum hardware. Local statistics, such as the degree distributions, remain accurate across scales with total variation distances ranging from $0.04$ to $0.20$, while global properties like strict bipartiteness degrade at the largest system sizes ($91$ and $153$ qubits). Notably, spectral bipartivity, a relaxation of strict bipartiteness, remains comparatively robust at higher qubit counts. These results establish practical baselines for the performance of shallow IQP circuits on current quantum hardware and demonstrate that, even without error mitigation, such circuits can learn and reproduce meaningful structural patterns in graph data, guiding future developments in quantum generative modeling for the NISQ era and beyond.

DOI
10.1103/jcdk-q3xc

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We introduce shallow instantaneous quantum polynomial-time (IQP) circuits as generative graph models, using an edge-qubit encoding to map graphs onto quantum states. Focusing on bipartite and Erdős-Rényi distributions, we study their expressivity and robustness through simulations and large-scale experiments. Noiseless simulations of 282828 qubits (888-node graphs) reveal that shallow IQP models can learn key structural features, such as the edge density and bipartite partitioning. On IBM's Aachen QPU, we scale experiments from 282828 to 153153153 qubits (888-181818 nodes) in order to characterize performance on real quantum hardware. Local statistics, such as the degree distributions, remain accurate across scales with total variation distances ranging from 0.040.040.04 to 0.200.200.20, while global properties like strict bipartiteness degrade at the largest system sizes (919191 and 153153153 qubits). Notably, spectral bipartivity, a relaxation of strict bipartiteness, remains comparatively robust at higher qubit counts. These results establish practical baselines for the performance of shallow IQP circuits on current quantum hardware and demonstrate that, even without error mitigation, such circuits can learn and reproduce meaningful structural patterns in graph data, guiding future developments in quantum generative modeling for the NISQ era and beyond.
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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.
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.
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.