Results 1 
7 of
7
Algorithms for Quantum Computation: Discrete Logarithms and Factoring
, 1994
"... A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a increase in computation time of at most a polynomial factor. It is not clear whether this is still true when quantum mechanics is taken into consi ..."
Abstract

Cited by 812 (7 self)
 Add to MetaCart
A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a increase in computation time of at most a polynomial factor. It is not clear whether this is still true when quantum mechanics is taken into consideration. Several researchers, starting with David Deutsch, have developed models for quantum mechanical computers and have investigated their computational properties. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems. (We thus give the first examples of quantum cryptanalysis.) 1 Introduction Since the discovery of quantum mechanics, people have found the behavior of...
unknown title
"... In recent years, much progress has been made in the study of quantum computation [1,2]. The first algorithm ..."
Abstract
 Add to MetaCart
In recent years, much progress has been made in the study of quantum computation [1,2]. The first algorithm
Quantum Algorithms for Graph Problems  A Survey
, 2006
"... In this survey we give an overview about important methods to construct quantum algorithms and quantum lower bounds for graph problems. We show how to use these methods, and we give a summary about the quantum complexity of the most important graph problems. At the end of our paper, we give some int ..."
Abstract
 Add to MetaCart
In this survey we give an overview about important methods to construct quantum algorithms and quantum lower bounds for graph problems. We show how to use these methods, and we give a summary about the quantum complexity of the most important graph problems. At the end of our paper, we give some interesting questions in this research area.
An application of the DeutschJosza algorithm to formal languages and the word problem in groups
, 2008
"... We adapt the DeutschJosza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original al ..."
Abstract
 Add to MetaCart
We adapt the DeutschJosza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for the word problem in groups with a particular kind of presentation. 1 The DeutschJosza algorithm and formal languages We apply a direct generalization of the DeutschJosza algorithm to the context of formal language theory. More particularly, we extend the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. For background information, we refer the reader to [1] and [2]. The DeutschJosza algorithm concerns maps f: {0,1} n − → {0,1}, which we may think of as words of length 2 n in a twoletter alphabet. Instead, let us consider words of length 2 n in a fourletter alphabet A = {a,b,c,d}. We identify the letters with binary strings of length 2: a ↔ 00, b ↔ 01, c ↔ 10 and d ↔ 11. Then words of length 2 n are in onetoone correspondence with maps f: {0,1} n − → {00,01,10,11}.
NP in BQP by Partial Measurement
, 1998
"... Abstract: By allowing measurement (readout) of some, but not all, qubits, this paper shows that NPcomplete problems are efficiently solvable by quantum computers in polynomial time with bounded probability (NP in BQP). This is demonstrated by describing a polynomially large network of quantum gate ..."
Abstract
 Add to MetaCart
Abstract: By allowing measurement (readout) of some, but not all, qubits, this paper shows that NPcomplete problems are efficiently solvable by quantum computers in polynomial time with bounded probability (NP in BQP). This is demonstrated by describing a polynomially large network of quantum gates that solves the 3SAT problem with bounded probability in polynomial time. The “weak ChurchTuring thesis ” is thus shown to be false. 1.
NP in BQP with Nonlinearity
, 1998
"... Abstract: If one modifies the laws of Quantum Mechanics to allow nonlinear evolution of quantum states, this paper shows that NPcomplete problems would be efficiently solvable in polynomial time with bounded probability (NP in BQP). With that (admittedly very unlikely) assumption, this is demonstra ..."
Abstract
 Add to MetaCart
Abstract: If one modifies the laws of Quantum Mechanics to allow nonlinear evolution of quantum states, this paper shows that NPcomplete problems would be efficiently solvable in polynomial time with bounded probability (NP in BQP). With that (admittedly very unlikely) assumption, this is demonstrated by describing a polynomially large network of quantum gates that solves the 3SAT problem with bounded probability in polynomial time. As in a previous paper by Abrams and Lloyd (but by a somewhat simpler argument), allowing nonlinearity in the laws of Quantum Mechanics would prove the “weak ChurchTuring thesis ” to be false. General Relativity is suggested as a possible mechanism to supply the necessary nonlinearity. 1.