Quantum embedding of graphs for subgraph counting

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 2 N working qubits and 2 ancilla qubits using its adjacency list, with worst-case gate complexity O(N2), 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.

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