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IEEE International Conference on Quantum Computing and Engineering (QCE)

Generative Graph Modeling on NISQ Computers

Oriol Balló-Gimbernat, Marcos Arroyo-Sánchez

Published Aug 30, 2025• 0 citations
Publisher

Abstract

Quantum generative models offer an inherently efficient sampling procedure, yet their training on quantum hardware is often a bottleneck. Parametrized Instantaneous Quantum Polynomial-time (IQP) circuits present a promising approach combining classical training with single-shot measurements for sample generation. However, its practical viability on Noisy Intermediate-Scale Quantum (NISQ) hardware remains an open question. In this work, we provide the first large-scale experimental validation of this method for learning graph distributions with both local (edge probabilities) and global (‘2-colorability’) constraints. While our graph encoding requires a number of qubits that scales quadratically with node count, our experiments, ranging from 28 to 153 qubits, demonstrate the approach's capabilities and limitations. For instance, a 153-qubit model successfully learned sparse ErdősRényi graphs with an average edge probability error below two percentage points. In contrast, learning the global property of bipartiteness proved more challenging and sensitive to noise, achieving 25.78 % accuracy for 18-node sparse instances. These results validate parametrized IQP circuits as a practical tool for generative modeling on near-term devices, while clearly delineating the critical challenges that must be addressed for future progress.

DOI
10.1109/qce65121.2025.10391

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Quantum generative models offer an inherently efficient sampling procedure, yet their training on quantum hardware is often a bottleneck. Parametrized Instantaneous Quantum Polynomial-time (IQP) circuits present a promising approach combining classical training with single-shot measurements for sample generation. However, its practical viability on Noisy Intermediate-Scale Quantum (NISQ) hardware remains an open question. In this work, we provide the first large-scale experimental validation of this method for learning graph distributions with both local (edge probabilities) and global (‘2-colorability’) constraints. While our graph encoding requires a number of qubits that scales quadratically with node count, our experiments, ranging from 28 to 153 qubits, demonstrate the approach's capabilities and limitations. For instance, a 153-qubit model successfully learned sparse ErdősRényi graphs with an average edge probability error below two percentage points. In contrast, learning the global property of bipartiteness proved more challenging and sensitive to noise, achieving 25.78 % accuracy for 18-node sparse instances. These results validate parametrized IQP circuits as a practical tool for generative modeling on near-term devices, while clearly delineating the critical challenges that must be addressed for future progress.
Generative modeling is one of the most promising applications of quantum machine learning, yet training and deploying Quantum Generative Models (QGMs) on near-term hardware remains effectively intractable due to prohibitive gradient estimation and implementation costs. We propose a resource-efficient approach based on shallow Instantaneous Quantum Polynomial-time (IQP) circuits that circumvents these bottlenecks by leveraging efficient classical training while retaining the guarantee of sampling hardness. To validate this approach, we formalize graph generation as a hierarchy of physical correlations, allowing us to map abstract data features, such as edge density and bipartiteness, directly to the quantum observables required to learn them. We validate our protocol through demonstrations both on real hardware (from 282828 to 153153153 qubits) and simulations (282828 qubits). Results show that while global structural features exhibit significant degradation beyond 919191 qubits, our models achieve high-precision reproduction of local correlations, even up to 153153153 qubits. These findings establish shallow IQP circuits as a robust, scalable candidate for generative tasks on current Noisy Intermediate-Scale Quantum (NISQ) devices.
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.
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.
Graph neural networks (GNNs) are a powerful framework for learning representations from graph-structured data, but their direct implementation on near-term quantum hardware remains challenging due to circuit depth, multi-qubit interactions, and qubit scalability constraints. In this work, we introduce a hybrid quantum graph learning architecture designed explicitly for unsupervised learning in the noisy intermediate-scale quantum (NISQ) regime. Our approach combines a variational quantum feature extraction layer with an edge-local and qubit-efficient quantum message-passing mechanism inspired by the Quantum Alternating Operator Ansatz (QAOA) framework. The message-passing operation is decomposed into pairwise interactions along graph edges using standard single- and two-qubit gates. For a graph with NNN nodes and nnn-qubit feature registers, this reduces the number of qubits required at one time from NnNnNn to at most 2n2n2n. We train the model using the Deep Graph Infomax objective to perform unsupervised node representation learning. The external class labels are not used during graph construction or training and are used only to evaluate the learned embeddings. Experiments on the Cora citation network and the Phase 3 release of the 1000 Genomes Project show that the quantum edge interaction contributes to the quality of the learned node representations.
The challenge to scaling quantum generative models on near-term hardware is training. Variational circuit Born machines require repeated quantum sampling and are prone to barren plateaus. Instantaneous Quantum Polynomial-time (IQP) Born machines sidestep both, since their loss is built from low-order Pauli-Z correlators that admit an unbiased classical estimator, while sampling from worst-case circuits in the class is conjectured to be classically hard. We take this train-on-classical, deploy-on-quantum workflow to a real high-energy-physics generative task, learning calorimeter shower profiles at 64 qubits and running the trained model on an IBM Heron r2 superconducting processor at 67 physical qubits. Three ingredients make it work. A uniform mixture of IQP circuits (MoIQP) widens the model class at single-circuit training cost. The Pearson-Stabilized Correlation Kernel (PSCK) biases descent toward the pairwise correlations that carry the shower-development physics, which the standard heat kernel systematically compresses. An exact deferred-measurement compilation collapses the mixture into a single IQP circuit, realized on hardware as a constant-depth dynamic circuit with zero SWAP insertions on the device's native heavy-hex graph. The trained model reconstructs the correlation structure to within 0.016 of the floor imposed by the encoding itself. Raw device samples reproduce the per-cell energy spectra and the full pairwise correlation structure at Pearson r = 0.989, up to a single global amplitude compression of depolarizing origin. A Gaussian copula fitted to the same training split matches the pairwise target more accurately than the quantum model at negligible cost. The contribution is therefore the classical trainability, exact compilation, and hardware deployability of a quantum generative model at this scale, not superiority over classical surrogates.
Quantum circuit Born machines (QCBMs) built with instantaneous quantum polynomial (IQP) circuits have loss functions that can be efficiently estimated classically, but require quantum computers during inference. Thus, classical computers can train large IQP QCBMs that exceed the qubit numbers of available quantum hardware. This motivates research into how qubit-limited quantum computers can be used to sample from larger IQP QCBMs. Leveraging insights from classical simulation of noisy IQP circuits, we propose a deployment protocol for IQP QCBMs that injects noise at the logical level to probabilistically decompose the original circuit into smaller clusters that can be sampled on qubit-limited quantum processors. We also apply lightweight resampling of known noisy qubits to mitigate the impact of the introduced error. Numerical simulations of our protocol applied to shallow IQP QCBMs and IBM Heron experiments deploying a 316-qubit IQP circuit onto 156-qubit hardware show that mitigation improves the similarity of the empirical distribution to the target distribution. The proposed deployment protocol also crucially preserves correlation structures more accurately than a classical baseline that has accurate single-qubit marginals.