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

• URL: http://arxiv.org/abs/2607.09469v1
• Date: Fri, 10 Jul 2026 14:42:55 GMT
• Title: A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity
• Abstract summary: We introduce a new notion of distance between two graph states $|Grangle$ and $|G'rangle$ on the same set of qubits.<n>This distance is the minimum number of ancilla qubits in a graph state $|widehatGrangle$.<n>We show how this distance yields quantum network analogs of many graph edit-distance problems.
• Score: 1.9607843515179006
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $k=1$.

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