IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Quantum algorithms for intersection and proximity problems

Kunihiko Sadakane, Norito Sugawara, Takeshi Tokuyama

Published May 1, 2003• 0 citations

Abstract

We discuss applications of quantum computation to geometric data processing. Especially, we give efficient algorithms for intersection problems and proximity problems. Our algorithms are based on Brassard et al.'s amplitude amplification method, and analogous to Buhrman et al.'s algorithm for element distinctness. Revealing these applications is useful for classifying geometric problems, and also emphasizing potential usefulness of quantum computation in geometric data processing. Thus, the results will promote research and development of quantum computers and algorithms.

Related papers

Quantum Computation in Computational Geometry

Kunihiko Sadakane, Noriko SUGAWARA, Takeshi Tokuyama

We discuss applications of quantum computation to geometric data processing. These applications include problems on convex hulls, minimum enclosing balls, linear programming, and intersection problems. Technically, we apply well-known Grover’s algorithm (and its variants) combined with geometric algorithms, and no further knowledge of quantum computing is required. However, revealing these applications and emphasizing potential usefulness of quantum computation in geometric data processing will promote research and development of quantum computers and algorithms.

Interdisciplinary Information SciencesJan 1, 20029 citations

Quantum Algorithms for Element Distinctness

Harry Buhrman, Christoph Dürr, Mark Heiligman +4 more

We present several applications of quantum amplitude amplification for deciding whether all elements in the image of a given function are distinct, for finding an intersection of two sorted tables, and for finding a triangle in a graph. Our techniques generalize and improve those of Brassard, Hoyer, and Tapp [ACM SIGACT News, 28 (1997), pp. 14--19]. This shows that in the quantum world element distinctness is significantly easier than sorting, in contrast to the classical world.

arXiv:quant-ph/0007016SIAM Journal on ComputingJan 1, 2005101 citations

Quantum Computing Based Geometric Optimization

Jincheng Zhang

Quantum Computing Based Geometric Optimization explores the potential of quantum algorithms to revolutionize optimization problems, specifically focusing on achieving enhanced efficiency compared to classical methods. This paper delves into the theoretical underpinnings and potential practical applications of this approach, emphasizing the unique capabilities offered by quantum computation in manipulating geometric data and solving complex optimization challenges. We present a framework for designing quantum-enhanced optimization algorithms, highlighting the key mathematical concepts and potential benefits across diverse domains. The core claim of this research centers on the utilization of quantum mechanics to significantly accelerate optimization processes, leading to improvements in computational speed and solution quality. The paper will outline the fundamental principles of quantum geometric optimization, detailing the advantages over conventional algorithms and exploring specific implementation strategies.

Sep 3, 20260 citations

Quantum algorithms for element distinctness

Harry Buhrman, Christoph Dürr, Mark Heiligman +4 more

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp (1998), and imply an O(N/sup 3/4/ log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with /spl Theta/(N log N) classical complexity. We also prove a lower bound of /spl Omega/(/spl radic/N) comparisons for this problem and derive bounds for a number of related problems.

IEEE Conference on Computational Complexity (CCC)Nov 13, 200294 citations

Geometry-Based Quantum Algorithm Design

Jincheng Zhang

This paper investigates the application of geometric association principles to quantum algorithm design, proposing a novel approach that leverages the inherent relationships within geometric structures to enhance algorithm efficiency and stability. Traditional quantum algorithms frequently focus on numerical precision, yet this research explores the potential of utilizing geometric structures to create algorithms with distinct advantages in specific contexts. We outline a design methodology centered around identifying and exploiting these connections to optimize quantum circuit construction and ultimately improve the performance of quantum computations. The core mechanism of this study centers on the systematic application of geometric principles to guide the development of quantum algorithms, aiming for a more robust and efficient computational process. This work provides a foundational framework for exploring the potential of geometric association in quantum algorithm design.

Aug 31, 20260 citations

Quantum Geometry Adaptive Computation

Jincheng Zhang

Quantum Geometry Adaptive Computation is a novel approach to computation leveraging the unique properties of quantum geometry. This paper explores the potential of utilizing quantum geometric principles to develop highly efficient algorithms and computational paradigms. The core claim centers on the design of a self-adaptive computation algorithm predicated on quantum geometry, specifically aiming to enhance computational performance through the inherent advantages of this geometric framework. This work investigates the theoretical underpinnings and practical implementation of this approach, demonstrating its promise for advanced computational tasks. We present a preliminary study exploring the impact of quantum geometry on specific computational challenges, highlighting its potential to surpass conventional methods. The research will focus on outlining the core mechanism, potential applications, and preliminary results, emphasizing the advancements achieved through the application of quantum geometry.

Sep 4, 20260 citations