A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity

2026-07-10

AI Summary

This study addresses the identification of highly entangled clusters in graph states within quantum networks and the quantification of distances between such graph states. To this end, the authors propose a distance measure based on the minimum number of auxiliary qubits required to transform one graph state into another, and they characterize this distance via vertex-induced subgraphs, thereby establishing the first analogy between graph state distance and graph edit distance. Introducing the notion of rank integrity, they prove its equivalence—up to a constant factor—to auxiliary integrity and develop a combinatorial framework integrating graph theory, GF(2) matrix rank analysis, and parameterized complexity theory. Their main theoretical contributions include showing that the rank integrity problem is XP but W[1]-hard when parameterized by $k$, and presenting an exact algorithm with time complexity $O(n^6)$ for the case $k=1$.

Abstract

We introduce a new notion of distance between two graph states $|G\rangle$ and $|G'\rangle$ on the same set of qubits. This distance is the minimum number of ancilla qubits in a graph state $|\widehat{G}\rangle$ from which both $|G\rangle$ and $|G'\rangle$ can be ``easily prepared''. (When preparing graph states, we are only allowed to use one-qubit Clifford gates, one-qubit Pauli measurements, and classical communication.) We give a graphical description of this distance through the lens of vertex-minors. We then show how this distance yields quantum network analogs of many graph edit-distance problems. Using this framework, we develop classical algorithms for identifying the ``highly entangled clusters'' of a graph state $|G\rangle$. The ancilla integrity problem asks, given a graph $G$ and integer $k$, for the minimum -- over all graph states $|G'\rangle$ with distance at most $k$ from $|G\rangle$ -- of the maximum component size of $G'$. Up to a factor of $2$ in the number of ancilla qubits, this problem is equivalent to rank integrity, where the distance between $G$ and $G'$ is instead the minimum rank of the sum of their adjacency matrices over $\text{GF}(2)$. We prove that rank integrity is XP parameterized by $k$. We also prove the complementary hardness result that rank integrity is W[1]-hard in $k$. Finally, we give an explicit $\mathcal{O}(n^6)$-time algorithm for ancilla integrity when $G$ has $n$ vertices and $k=1$.

Problem

• graph states
• ancilla integrity
• rank integrity
• entanglement clustering
• parameterized complexity

Innovation

• graph states
• vertex-minors
• rank integrity
• parameterized complexity
• quantum network clustering