Circuit Synthesis of Marked Clique Problem using Quantum Walk
TL;DR: In this article, a quantum circuit implementation for the oracle of the marked clique problem based on Quantum Walk approach is proposed, which is the first approach in regards to the quantum circuit synthesis of the marking clique oracle in binary quantum domain.
Abstract: Many applications such as Element Distinctness, Triangle Finding, Boolean Satisfiability, Marked Subgraph problem can be solved by Quantum Walk, which is the quantum version of classical random walk without intermediate measurement. To design a quantum circuit for a given quantum algorithm, which involves Quantum Walk search, we need to define an oracle circuit specific to the given algorithm and the diffusion operator for amplification of the desired quantum state. In this paper, we propose a quantum circuit implementation for the oracle of the marked clique problem based on Quantum Walk approach. To the best of our knowledge, this is a first of its kind approach in regards to the quantum circuit synthesis of the marked clique oracle in binary quantum domain. We have performed the simulation of the proposed oracle circuit in Matlab.
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TL;DR: In this paper, the authors considered factoring integers and finding discrete logarithms on a quantum computer and gave an efficient randomized algorithm for these two problems, which takes a number of steps polynomial in the input size of the integer to be factored.
Abstract: A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
7,427 citations
01 Jul 1996
TL;DR: In this paper, it was shown that a quantum mechanical computer can solve integer factorization problem in a finite power of O(log n) time, where n is the number of elements in a given integer.
Abstract: were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------
6,335 citations
TL;DR: Two backtracking algorithms are presented, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique, and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed.
Abstract: Description bttroductian. A maximal complete subgraph (clique) is a complete subgraph that is not contained in any other complete subgraph. A recent paper [1] describes a number of techniques to find maximal complete subgraphs of a given undirected graph. In this paper, we present two backtracking algorithms, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique. The first version is a straightforward implementation of the basic algorithm. It is mainly presented to illustrate the method used. This version generates cliques in alphabetic (lexicographic) order. The second version is derived from the first and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed. This version tends to produce the larger cliques first and to generate sequentially cliques having a large common intersection. The detailed algorithm for version 2 is presented here. Description o f the algorithm--Version 1. Three sets play an important role in the algorithm. (1) The set compsub is the set to be extended by a new point or shrunk by one point on traveling along a branch of the backtracking tree. The points that are eligible to extend compsub, i.e. that are connected to all points in compsub, are collected recursively in the remaining two sets. (2) The set candidates is the set of all points that will in due time serve as an extension to the present configuration of compsub. (3) The set not is the set of all points that have at an earlier stage already served as an extension of the present configuration of compsub and are now explicitly excluded. The reason for maintaining this set trot will soon be made clear. The core of the algorithm consists of a recursively defined extension operator that will be applied to the three sets Just described. It has the duty to generate all extensions of the given configuration of compsub that it can make with the given set of candidates and that do not contain any of the points in not. To put it differently: all extensions of compsub containing any point in not have already been generated. The basic mechanism now consists of the following five steps:
2,405 citations
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TL;DR: The concept of quantum random walk is introduced, and it is shown that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding classical random walk.
Abstract: We introduce the concept of quantum random walk, and show that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding classical random walk A quantum-optics application is described
1,518 citations
TL;DR: A brief overview of quantum walks, with emphasis on their algorithmic applications, can be found in this article, where the authors describe quantum walks as quantum counterparts of Markov chains, and present several applications of quantum walk.
Abstract: Quantum walks are quantum counterparts of Markov chains. In this article, we give a brief overview of quantum walks, with emphasis on their algorithmic applications.
579 citations