Quantum embedding of graphs for subgraph counting

2026-04-20

AI Summary

This work proposes the first unified quantum embedding framework for counting subgraphs—such as triangles, cycles, and cliques—in graphs. The method encodes an N-vertex graph via its adjacency list into a quantum state and designs measurement operators tailored to the edge structure of the target subgraph. By performing measurements on the m-fold tensor product of this state, the algorithm estimates the number of occurrences of the subgraph. Requiring only 2⌈log₂N⌉ working qubits and two ancilla qubits, it achieves a worst-case gate complexity of O(N²) within quantum logspace, offering a significant advantage in space complexity over classical approaches. This result establishes a novel quantum paradigm for graph motif counting.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We develop a unified quantum framework for subgraph counting in graphs. We encode a graph on $N$ vertices into a quantum state on $2\lceil \log_2 N \rceil$ working qubits and $2$ ancilla qubits using its adjacency list, with worst-case gate complexity $O(N^2)$, which we refer to as the graph adjacency state. We design quantum measurement operators that capture the edge structure of a target subgraph, enabling estimation of its count via measurements on the $m$-fold tensor product of the adjacency state, where $m$ is the number of edges in the subgraph. We illustrate the framework for triangles, cycles, and cliques. This approach yields quantum logspace algorithms for motif counting, with no known classical counterpart.

Problem

• subgraph counting
• quantum embedding
• graph motif
• quantum algorithm
• adjacency state

Innovation

• quantum embedding
• subgraph counting
• graph adjacency state
• quantum logspace algorithm
• motif counting