Results 1  10
of
18
Quantum algorithms revisited
 Proceedings of the Royal Society of London A
, 1998
"... Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference. We use th ..."
Abstract

Cited by 146 (15 self)
 Add to MetaCart
Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision. 1.
The hidden subgroup problem and eigenvalue estimation on a quantum computer
 Lecture Notes in Computer Science
, 1999
"... Abstract. A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of e ..."
Abstract

Cited by 58 (2 self)
 Add to MetaCart
Abstract. A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits. 1
Entanglement and quantum computation
 The Geometric Universe
, 1998
"... The phenomenon of quantum entanglement is perhaps the most enigmatic feature of the formalism of quantum theory. It underlies many of the most curious and controversial aspects of the quantum mechanical description of the world. In [1] Penrose gives a delightful and accessible account of entanglemen ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
The phenomenon of quantum entanglement is perhaps the most enigmatic feature of the formalism of quantum theory. It underlies many of the most curious and controversial aspects of the quantum mechanical description of the world. In [1] Penrose gives a delightful and accessible account of entanglement illustrated by some
Quantum factoring, discrete logarithms and the hidden subgroup problem
"... Amongst the most remarkable successes of quantum computation are Shor’s efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential ingredients of these algorithms and draw out the unifying gener ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
Amongst the most remarkable successes of quantum computation are Shor’s efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential ingredients of these algorithms and draw out the unifying generalization of the socalled abelian hidden subgroup problem. This involves an unexpectedly harmonious alignment of the formalism of quantum physics with the elegant mathematical theory of group representations and fourier transforms on finite groups. Finally we consider the nonabelian hidden subgroup problem mentioning some open questions where future quantum algorithms may be expected to have a substantial impact. 1
Quantum algorithms: Entanglement enhanced information processing
 Phil. Trans. R. Soc. Lond. A
, 1998
"... Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algori ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the grouptheoretic formalism leads to the standard quantum network and identifying the property of entanglement that gives rise to the exponential speedup (compared to the classical FFT). Finally we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.
A Rosetta stone for quantum mechanics with an introduction to quantum computation
, 2002
"... Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading ..."
Abstract

Cited by 21 (11 self)
 Add to MetaCart
Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading
Searching in Grover’s Algorithm
"... Grover’s algorithm is usually described in terms of the iteration of a compound operator of the form Q = −HI0HIx0. Although it is quite straightforward to verify the algebra of the iteration, this gives little insight into why the algorithm works. What is the significance of the compound structure o ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Grover’s algorithm is usually described in terms of the iteration of a compound operator of the form Q = −HI0HIx0. Although it is quite straightforward to verify the algebra of the iteration, this gives little insight into why the algorithm works. What is the significance of the compound structure of Q? Why is there a minus sign? Later it was discovered that H could be replaced by essentially any unitary U. What is the freedom involved here? We give a description of Grover’s algorithm which provides some clarification of these questions.
Quantum algorithms: Entanglementenhanced information processing
 The Geometric Universe: Science, Geometry, and the Work of Roger Penrose
, 1998
"... We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform (FFT) algorith ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform (FFT) algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the grouptheoretic formalism leads to the standard quantum network, and identify the property of entanglement that gives rise to the exponential speedup (compared to the classical FFT). Finally, we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Continuous Quantum Hidden Subgroup Algorithms
, 2003
"... In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Pat ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L 2 paths x: [0, 1] → R n in real nspace R n. Also included is a curious discrete QHS algorithm which is dual to Shor’s algorithm. Contents 1