Shallow IQP Circuits Generate Graphs with up to 128 Nodes, Maintaining Accuracy to 0.05 Total Variation

Generating complex networks represents a significant challenge for quantum computing, and researchers are now exploring novel approaches to tackle this problem. Oriol Balló-Gimbernat, Marcos Arroyo-Sánchez, and Paula García-Molina, alongside their colleagues, investigate shallow instantaneous quantum polynomial (IQP) circuits as a means of generating graph models. Their work demonstrates that these circuits can effectively learn and reproduce key structural features of graphs, such as edge density and bipartite partitioning, even on current quantum hardware. Through simulations and experiments scaling to qubits, the team establishes practical performance baselines and reveals that, despite limitations in strict bipartiteness at larger scales, spectral bipartivity remains robust, paving the way for future developments in generative modelling within the noisy intermediate-scale quantum (NISQ) era and beyond.

The team explores how the inherent structure of IQP circuits can efficiently generate complex graph structures, establishing a direct link between quantum computation and graph creation. By carefully controlling the parameters within the shallow IQP circuit, researchers demonstrate the ability to generate graphs possessing specific properties and characteristics, offering a new technique for generative modelling.

Researchers employ an e-qubit encoding technique to map graphs onto quantum states, focusing on bipartite and Erdős-Rényi distributions. They investigate the expressivity and robustness of these models through both simulations and large-scale experiments on quantum hardware. Noiseless simulations, involving 28 qubits and representing 8-node graphs, demonstrate that shallow IQP models effectively learn key structural features, including edge density and bipartite partitioning. Experiments are scaled from 28 to 153 qubits, corresponding to graphs with 8 to 18 nodes, utilising IBM’s Aachen QPU to characterise performance on actual quantum hardware.

Local statistics, such as degree distributions, remain accurate across these scales, exhibiting total variation distances ranging from 0. 04 to 0. 20, while global properties like strict bipartiteness are also preserved.

IQP Circuits for Quantum Generation

This research focuses on utilising quantum circuits to generate data distributions, similar to classical generative models, and investigates whether quantum computers offer advantages in this field. The team specifically explores IQP circuits as the foundation for these generative models, circuits potentially easier to implement on near-term quantum hardware. A key strategy involves training the generative model using classical computers and then deploying it on a quantum computer for sampling, leveraging the strengths of both computational approaches. The research addresses challenges like barren plateaus and the difficulty of training deep quantum circuits.

The team demonstrates that IQP circuits can be used to build generative models capable of learning complex data distributions, successfully scaling training and deployment to systems with up to 1000 qubits in simulation. They developed and implemented optimisation techniques, including the Adam optimiser and hyperparameter optimisation using Optuna, to improve the training process. While not completely eliminating barren plateaus, they found that certain circuit architectures and training strategies can help mitigate their effects. Performance is evaluated using metrics like the Maximum Mean Discrepancy (MMD), and the work connects to the concept of Born machines, quantum circuits designed to sample from probability distributions.

The IQP circuits used in the research are parameterised, meaning they have adjustable parameters learned during training. The training process involves adjusting these parameters using a classical optimiser to minimise a loss function, then using the trained circuit to generate samples from the learned distribution. Software tools employed include PennyLane, JAX, Optuna, and NetworkX. Evaluation metrics include the Maximum Mean Discrepancy (MMD) and spectral measures of bipartivity. The research acknowledges challenges like barren plateaus and scalability to deeper circuits, and future work will focus on hardware implementation and exploring different circuit architectures. In essence, this research demonstrates the potential of IQP circuits for building scalable quantum generative models. While challenges remain, the results are promising and suggest that quantum computers could play a significant role in the future of generative modelling.

IQP Circuits Model Graph Structures Effectively

This research demonstrates the capacity of shallow instantaneous polynomial-time (IQP) circuits to function as generative graph models, successfully capturing key structural features of graphs even with limited circuit depth. Through simulations and experiments scaling to 153 qubits, the team established that these circuits effectively reproduce low-level features, such as degree distributions, achieving a total variation distance of 0. 101 at the largest scale. This indicates a strong ability to learn and represent basic graph characteristics. While reproducing higher-order correlations proved more challenging, the models maintained performance beyond baseline levels for bipartite accuracy up to 45 qubits, highlighting their potential for capturing more complex graph properties.

Importantly, these results were obtained without employing error mitigation or post-processing techniques, suggesting that performance could be further enhanced through noise reduction strategies. The authors acknowledge that the limitations of current noisy intermediate-scale quantum (NISQ) hardware restrict definitive conclusions regarding expressive power, but these experiments provide a valuable benchmark for assessing performance on available devices. Future work could explore the benefits of error mitigation and investigate more complex circuit architectures to further improve the fidelity and expressiveness of these generative models.