相似子图识别下的固定边移除量子搜索算法
Quantum search algorithm for similar subgraph identification under fixed edge removal
AI总结：
本文提出一种量子算法，通过将子图拓扑表示为Dicke态，结合幅值估计和最小值寻找，实现对子图拉普拉斯矩阵的快速优化，相比经典方法有多项式加速。
摘要：
我们介绍了一种新型量子算法，用于相似子图识别，形式为NP难的卡数约束二元二次优化问题。给定具有拉普拉斯矩阵B的加权参考图，我们的算法确定在相同顶点集上具有拉普拉斯矩阵B'的子图，其中x条边处于非活动状态，最小化Frobenius距离||B - B'||_F^2。我们将C(N,x)种图拓扑表示为等权重叠加的Dicke态，使量子态与参考图向量化拉普拉斯矩阵相关联。结合幅值估计和最小值寻找方法，该算法相比经典暴力搜索算法有多项式加速O(√(N^x/x!)*N log log N)。我们在标准测试用例上展示了该方法的应用，这些用例代表电力网络，通过测量重建||B - B'||_F^2，并展示如何进一步利用该方法计算与给定向量相关的能量泛函如拉普拉斯矩阵的二次形式。
英文摘要：
We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian $\boldsymbol{B}$, our algorithm determines the subgraph featuring Laplacian $\boldsymbol{B'}$ on the same vertex set, but $x$ out of $N$ inactive edges, minimizing the Frobenius distance $||\boldsymbol{B} - \boldsymbol{B'}||_\mathrm{F}^2$. We represent the $\binom{N}{x}$ graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up $\mathcal{O}(\sqrt{N^{x}/x!}N\log\log N)$ compared to $\mathcal{O}(N^{x+1}/x!)$ of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing $||\boldsymbol{B} -\boldsymbol{B'}||_\mathrm{F}^2$ from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
