arXiv 2102.092052021-02-18quant-ph

三能级量子元量子态的量子退火成簇

Clustering by quantum annealing on three-level quantum elements qutrits

V. E. Zobov, I. S. Pichkovskiy

备注：: 11 pages, 4 figures

摘要：

聚类是根据某些属性的接近程度对数据进行分组。我们报告的可能性，增加了聚类的点在一个平面上使用人工量子神经网络的两级神经元称为量子比特由自旋S = 1/2表示的三个级别的神经元称为量子比特由自旋S = 1表示的替代后的效率。这个问题已经解决了缓慢的绝热变化的哈密顿量的时间。提出了用投影算子控制Qutrit系统的方法，并进行了数值模拟。建立了两种著名聚类方法--one-hot编码和k-means ++的哈密顿量。第一种方法已被用来划分一组六个点到三个或两个集群和第二种方法，划分一组九个点到三个集群和七个点到四个集群。模拟结果表明，在自旋S = 1的量子数上，聚类问题可以得到有效的解决.证明了在量子位上成簇比在量子位上成簇的优势。特别地，表示数据点所需的量子位数比量子比特数小log 2 N /log 3 N的因子。由于对于qutrits，将数据点划分为三个簇比两个簇更容易，因此加速了将数据划分为大量簇的近似分层过程。在将数据精确划分为三个以上的簇时，提出了用对应的多自旋子系统的状态数来对簇进行编号，而不是用单个自旋的数目。这甚至减少了实现算法所需的qutrits的数量（N log 3 K而不是NK）。

英文摘要：

Clustering is grouping of data by the proximity of some properties. We report on the possibility of increasing the efficiency of clustering of points in a plane using artificial quantum neural networks after the replacement of the two-level neurons called qubits represented by the spins S = 1/2 by the three-level neurons called qutrits represented by the spins S = 1. The problem has been solved by the slow adiabatic change of the Hamiltonian in time. The methods for controlling a qutrit system using projection operators have been developed and the numerical simulation has been performed. The Hamiltonians for two well-known cluster-ing methods, one-hot encoding and k-means ++, have been built. The first method has been used to partition a set of six points into three or two clusters and the second method, to partition a set of nine points into three clusters and seven points into four clusters. The simulation has shown that the clustering problem can be ef-fectively solved on qutrits represented by the spins S = 1. The advantages of clustering on qutrits over that on qubits have been demonstrated. In particular, the number of qutrits required to represent data points is smaller than the number of qubits by a factor of log2 N / log3 N . Since, for qutrits, it is easier to partition the data points into three clusters rather than two ones, the approximate hierarchical procedure of data partition-ing into a larger number of clusters is accelerated. At the exact data partition into more than three clusters, it has been proposed to number the clusters by the numbers of states of the corresponding multi-spin subsys-tems, instead of using the numbers of individual spins. This reduces even more the number of qutrits (N log3 K instead of NK ) required to implement the algorithm.

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。

摘要：

英文摘要：

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