Shallow IQP circuits and graph generation

TL;DR

This work investigates shallow IQP circuit Born Machines as generative models for graphs, encoding undirected graphs via an edge-qubit mapping and training with a classical MMD objective while sampling on IBM's Aachen QPU to probe hardware performance. The authors analyze two graph families, Erdős–Rényi and bipartite, and quantify learning via density, degree distribution, and bipartiteness-related features, including a spectral bipartivity measure. They demonstrate that local, low-bodied features are learnable and robust to noise up to 153 qubits, whereas global binary features degrade with scale, providing a practical baseline for NISQ-era quantum graph generation. The results highlight the potential of IQP-based generative modeling on near-term devices, while outlining challenges and future directions such as error mitigation and alternative training schemes to exploit quantum advantages in structured data modeling.

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.

Figures (7)

• Figure 1: Overview of the proposed workflow. Top: Model training is performed entirely on classical hardware via a classical estimation of the MMD loss. Bottom: Sampling is carried out on IBM’s Aachen quantum processor.
• Figure 2: Top: Adjacency matrix representation of a $7-$node undirected graph. Bottom: Corresponding bit string encoding constructed from the upper-triangular part of the adjacency matrix.
• Figure 3: Circuit diagram of a parameterized IQP circuit. The unitary $D_Z(\boldsymbol{\theta})$ contains all diagonal parameterized gates. For constant-depth $D_Z(\boldsymbol{\theta})$, the circuit realizes a shallow instance of the IQP model.
• Figure 4: Comparison of graph generation metrics between NISQ hardware and classical simulations for identical models. Panels show (a) generated density $\mathbb{E}[\rho]$, (b) bipartite accuracy, (c) spectral bipartivity $\mathbb{E}[\beta]$, (d) MMD$(q_\theta,p)$. The red dashed line denotes perfect agreement. Deviations highlight discrepancies due to hardware noise and finite sampling.
• Figure 5: Degree distributions obtained from models from $28$ to $153$ qubits trained on Erdős-Rényi datasets and executed on NISQ hardware. Bars indicate empirical node-degree frequencies, while solid lines denote theoretical binomial targets. The total variation distance (TVD) measures the deviation between generated and target distributions.

Theorems & Definitions (3)

• Definition 1: $k$-bodied feature
• Definition 2: Global feature
• Definition 3