arXiv subjects

Kimikazu Kato

Publications and source records attributed to Kimikazu Kato.

6 recordsLinked to original sources

arXiv2010

Solving $k$-Nearest Neighbor Problem on Multiple Graphics Processors

The recommendation system is a software system to predict customers' unknown preferences from known preferences. In the recommendation system, customers' preferences are encoded into vectors, and finding the nearest vectors to each vector is an essential part. This vector-searching part of the problem is called a $k$-nearest neighbor problem. We give an effective algorithm to solve this problem on multiple graphics processor units (GPUs). Our algorithm consists of two parts: an $N$-body problem and a partial sort. For a algorithm of the $N$-body problem, we applied the idea of a known algorithm for the $N$-body problem in physics, although another trick is need to overcome the problem of small sized shared memory. For the partial sort, we give a novel GPU algorithm which is effective for small $k$. In our partial sort algorithm, a heap is accessed in parallel by threads with a low cost of synchronization. Both of these two parts of our algorithm utilize maximal power of coalesced memory access, so that a full bandwidth is achieved. By an experiment, we show that when the size of the problem is large, an implementation of the algorithm on two GPUs runs more than 330 times faster than a single core implementation on a latest CPU. We also show that our algorithm scales well with respect to the number of GPUs.

cs.IR↗

arXiv2008

Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity

In quantum information theory, a geometric approach, known as "quantum information geometry," has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel.

quant-ph↗

arXiv2007

Degeneracy of Angular Voronoi Diagram

Angular Voronoi diagram was introduced by Asano et al. as fundamental research for a mesh generation. In an angular Voronoi diagram, the edges are curves of degree three. From view of computational robustness we need to treat the curves carefully, because they might have a singularity. We enumerate all the possible types of curves that appear as an edge of an angular Voronoi diagram, which tells us what kind of degeneracy is possible and tells us necessity of considering a singularity for computational robustness.

cs.CG↗

arXiv2006

Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity

We give a new geometric interpretation of quantum pure states. Using Voronoi diagrams, we reinterpret the structure of the space of pure states as a subspace of the quantum state space. In addition to the known coincidence of some Voronoi diagrams for one-qubit pure states, we will show that even for mixed one-qubit states, as far as sites are given as pure states, the Voronoi diagram with respect to some distances -- the divergence, the Bures distance, and the Euclidean distance -- are all the same. As to higher level pure quantum states, for the divergence, the Fubini-Study distance, and the Bures distance, the coincidence of the diagrams still holds, while the coincidence of the diagrams with respect to the divergence and the Euclidean distance no longer holds. That fact has a significant meaning when we try to apply the method used for a numerical estimation of a one-qubit quantum channel capacity to a higher level system.

quant-ph↗

arXiv2006

On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation

For one-qubit pure quantum states, it is already proved that the Voronoi diagrams with respect to two distances -- Euclidean distance and the quantum divergence -- coincide. This fact is a support for a known method to calculate the Holevo capacity. To consider an applicability of this method to quantum states of a higher level system, it is essential to check if the coincidence of the Voronoi diagrams also occurs. In this paper, we show a negative result for that expectation. In other words, we mathematically prove that those diagrams no longer coincide in a higher dimension. That indicates that the method used in one-qubit case to calculate the Holevo capacity might not be effective in a higher dimension.

quant-ph↗

arXiv2006

Voronoi Diagrams for Pure 1-qubit Quantum States

1-qubit quantum states form a space called the three-dimensional Bloch ball. To compute Holevo capacity, Voronoi diagrams in the Bloch ball with respect to the quantum divergence have been used as a powerful tool. These diagrams basically treat mixed quantum states corresponding to points in the interior of the Bloch ball. Due to the existence of logarithm in the quantum divergence, the diagrams are not defined on pure quantum states corresponding to points on the two-dimensional sphere. This paper first defines the Voronoi diagrams for pure quantum states on the Bloch sphere by the Fubini-Study distance and the Bures distance. We also introduce other Voronoi diagrams on the sphere obtained by taking a limit of Voronoi diagrams for mixed quantum states by the quantum divergences in the Bloch ball. These diagrams are shown to be equivalent to the ordinary Voronoi diagram on the sphere.

quant-ph↗