具有变分回路的Laplacian本征映射：图数据的量子嵌入
Laplacian Eigenmaps with variational circuits: a quantum embedding of graph data
摘要：
随着量子算法的发展，人们正在仔细研究高成本计算，以期获得量子优势。虽然图为多个现实世界的问题提供了一个方便的框架，但它们的分析仍然具有高计算量和空间。通过将图数据映射到一个低维空间，在这个空间中，图的结构信息被保留，拉普拉斯矩阵的特征向量构成了一个强大的节点嵌入，称为拉普拉斯特征映射。计算这些嵌入本身就是一项昂贵的任务，因为使用特定的稀疏方法，拉普拉斯矩阵的特征分解的成本为O（$rn^2$），$r$是非零元素的比率。提出了一种利用量子变分电路计算拉普拉斯本征映射的方法。我们的算法的思想是通过采用变分量子本征求解器算法来得到拉普拉斯矩阵的本征态，该矩阵可以被认为是一个哈密顿算符。通过同时估计拉普拉斯算子的$d$第一特征向量，我们的算法直接生成图的一个$d$维数量子嵌入。我们证明了通过在其上实现量子分类器来将嵌入用于图机器学习任务是可能的。整个电路包括一个完整的量子节点分类算法。利用量子模拟器对32节点图进行了测试，结果表明该算法可以达到与经典的拉普拉斯特征映射算法相似的性能。虽然这种近似方法的数学性质还没有完全理解，但这种算法为使用噪声量子计算机进行图形预处理开辟了前景。
英文摘要：
With the development of quantum algorithms, high-cost computations are being scrutinized in the hope of a quantum advantage. While graphs offer a convenient framework for multiple real-world problems, their analytics still comes with high computation and space. By mapping the graph data into a low dimensional space, in which graph structural information is preserved, the eigenvectors of the Laplacian matrix constitute a powerful node embedding, called Laplacian Eigenmaps. Computing these embeddings is on its own an expensive task knowing that using specific sparse methods, the eigendecomposition of a Laplacian matrix has a cost of O($rn^2$), $r$ being the ratio of nonzero elements. We propose a method to compute a Laplacian Eigenmap using a quantum variational circuit. The idea of our algorithm is to reach the eigenstates of the laplacian matrix, which can be considered as a hamiltonian operator, by adapting the variational quantum eigensolver algorithm. By estimating the $d$ first eigenvectors of the Laplacian at the same time, our algorithm directly generates a $d$ dimension quantum embedding of the graph. We demonstrate that it is possible to use the embedding for graph machine learning tasks by implementing a quantum classifier on the top of it. The overall circuit consists in a full quantum node classification algorithm. Tests on 32 nodes graph with a quantum simulator shows that we can achieve similar performances as the classical laplacian eigenmap algorithm. Although mathematical properties of this approximate approach are not fully understood, this algorithm opens perspectives for graph pre-processing using noisy quantum computers.
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