AI 中文总结
研究同一组量子比特上两个图态距离的新概念，通过顶点子式图形描述，产生图编辑距离问题的量子网络类似物，开发经典算法识别高度纠缠簇，证明秩完整性关于\(k\)的参数化复杂度及难度，并给出特定情况下辅助完整性算法。
AI 中文摘要
我们引入了在同一组量子比特上两个图态\(|G\rangle\)和\(|G'\rangle\)之间距离的新概念。此距离是图态\(|\widehat{G}\rangle\)中辅助量子比特的最小数量，从该图态可‘轻松制备’\(|G\rangle\)和\(|G'\rangle\)。（制备图态时，仅允许使用单比特克利福德门、单比特泡利测量和经典通信。）我们通过顶点子式对该距离进行图形描述。接着展示此距离如何产生许多图编辑距离问题的量子网络类似物。利用此框架，我们开发经典算法来识别图态\(|G\rangle\)的‘高度纠缠簇’辅助完整性问题是，给定图\(G\)和整数\(k\)，求与\(|G\rangle\)距离至多为\(k\)的所有图态\(|G'\rangle\)中\(G'\)最大连通分量大小的最小值。在辅助量子比特数量上相差因子\(2\)的情况下，此问题等同于秩完整性，其中\(G\)与\(G'\)之间的距离是它们邻接矩阵在\(\text{GF}(2)\)上求和的最小秩。我们证明秩完整性关于\(k\)是XP参数化的。还证明了互补的难度结果，即秩完整性关于\(k\)是W[1] - 难的。最后给出当\(G\)有\(n\)个顶点且\(k = 1\)时辅助完整性的显式\(\mathcal{O}(n^6)\)时间算法。
英文摘要
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 $k=1$.
