## Quantum random walks - an introductory overview (2003)

Venue: | Contemporary Physics |

Citations: | 76 - 2 self |

### BibTeX

@ARTICLE{Kempe03quantumrandom,

author = {J. Kempe and Albert Einstein},

title = {Quantum random walks - an introductory overview},

journal = {Contemporary Physics},

year = {2003},

volume = {44},

pages = {0303081}

}

### Years of Citing Articles

### OpenURL

### Abstract

This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions. 1. Overview Ever since the discovery of quantum mechanics people have been puzzled by the counter-intuitive character of the laws of nature. Over time we have learned to accept more and more effects that are unimaginable in a classical Newtonian world. Modern technology exploits quantum effects both to our benefit and detriment—among the memorable examples we should cite laser technology and not omit the atomic bomb. In recent years interest in quantum information theory has been generated by the prospect of employing its laws to design devices of surprising power [1]. New ideas include quantum cryptography [2, 3] and quantum computation. In 1994 Shor [4] discovered a quantum algorithm to factor numbers efficiently (that is in time that grows only polynomically with the length of the number to be factored). This has unleashed a wave of activity across a broad range of disciplines: physics, computer science, mathematics and engineering. This fruitful axis of research has uncovered many new effects that are strikingly different from their classical counterparts, both from the physical point of view as well as from a computer science and communication theory perspective. Over time these communities have gained a greater understanding of the concepts and notions of the other. The idea that information cannot be separated from

### Citations

2343 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ... investigate the importance of random walks for algorithms in computer science. 4. Random walks in computer science Classical random walks form one of the cornerstones of theoretical computer science =-=[30, 31]-=-. As algorithmic tools, they have been applied to a variety of central problems [32], such as estimation of the volume of a convex body [33], approximation of the permanent of a matrix [34], and disco... |

1879 |
An Introduction to Probability Theory and its Applications
- Feller
- 1967
(Show Context)
Citation Context ...ase the Hamming distance to some true assignment is at least 1/2, but can be larger.s320 J. Kempe 4.2. Classical random walks Let us state without proofs some known facts about random walks on graphs =-=[36]-=-. A simple random walk on an undirected graph, G(V, E), is described by repeated applications of a stochastic matrix M, where Mi,j =1/diif (i, j) is an edge in G and di the degree of i (we use the not... |

1872 | Randomized Algorithms
- Motwani
- 1995
(Show Context)
Citation Context ... investigate the importance of random walks for algorithms in computer science. 4. Random walks in computer science Classical random walks form one of the cornerstones of theoretical computer science =-=[30, 31]-=-. As algorithmic tools, they have been applied to a variety of central problems [32], such as estimation of the volume of a convex body [33], approximation of the permanent of a matrix [34], and disco... |

373 | Reversible markov chains and random walks on graphs (monograph in preparation),” http://statwww.berkeley.edu/users/aldous/RWG/book.html - Aldous, Fill |

180 |
Algorithms for Random Generation and Counting: A Markov Chain Approach
- Sinclair
- 1993
(Show Context)
Citation Context ...m walks in computer science Classical random walks form one of the cornerstones of theoretical computer science [30, 31]. As algorithmic tools, they have been applied to a variety of central problems =-=[32]-=-, such as estimation of the volume of a convex body [33], approximation of the permanent of a matrix [34], and discovery of satisfying assignments for Boolean formulae [35]. They provide a general par... |

119 |
Quantum Optics
- Scully, Zubairy
- 1997
(Show Context)
Citation Context ...gle cavity mode, whereas the coin states are the internal atomic states{. Here a random walk on the circle is achieved. Let us sketch the scheme of [43] in more detail (for the optical background see =-=[47]-=-). The field mode of the cavity is described by the (infinite) Hilbert space of a harmonic oscillator spanned by photon number states {jni : n = 0,1,. . .}. Let ~N be the number operator acting as ^Nj... |

10 | private communication - Preskill |

3 |
eprint quant-ph/0207008
- Bach, Coppersmith, et al.
(Show Context)
Citation Context ...nd drops off exponentially several standard deviations s away from the origin. To uncover more striking differences of the quantum walk we can place it on a bounded line, either one-sided or twosided =-=[12, 16]-=-. In other words we can insert one or two absorbing boundaries on the line. Formally an absorbing boundary in position jbi corresponds to a partial measurement of the process at every time step. More ... |

3 |
e-print quant-ph/0209131
- Childs, Cleve, et al.
(Show Context)
Citation Context ...replacing all diagonal entries of 2 by 3 as seen in figure 13. Solving this modified problem is the lore of a quantum physicist and the solution can readily be given in terms of Bessel functions (see =-=[27]-=- for a much more complete treatment). It can be seen that the speed of propagation of the random walk on the infinite homogeneous line is linear in the time T. In other words there are constants a 5 b... |

1 |
unpublished. random walks: an introductory overview
- Watrous
- 2000
(Show Context)
Citation Context ...n [6] with the discretization of the Dirac equation in mind. In the era of quantum information it was rediscovered in works by Meyer [7, 8] in connection with quantum cellular automata and by Watrous =-=[9]-=- in the context of halting of quantum Turing machines and in a slightly modified version (with measurements) in [10] thinking about spacebounded quantum computation. As a possible computational tool a... |

1 |
2002, Proceedins of RANDOM 2002, edited by
- Moore, Russell
(Show Context)
Citation Context ...lk when we are interested in questions like the speed of propagation of the random walk to the opposite corner. If we wish to retain this symmetry property for the quantum random walk it can be shown =-=[17]-=- that the only coins that are unitary and permutation symmetric are of the following form 8 9 a b b ... b b b a b ... b b b b a ... b b Ga;b ; ð23Þ ... >: b b b ... a b>; b b b ... b a 2 where a and... |

1 | 2002, Unconventional Models of Computation - Yamasaki, Kobayashi, et al. |

1 |
arXive eprint quant-ph/0207028
- Sanders, Bartlett, et al.
- 2002
(Show Context)
Citation Context ... the random walk{. If nothing else the variance of the quantum random walk so obtained can serve as a benchmarking protocol for ion trap quantum computers. Another recent suggestion by Sanders et al. =-=[43]-=- puts the random walk into a quantum electrodynamics device. The physical system is an optical cavity traversed by an atom. During the traversal the internal electronic levels of the atom interact wit... |

1 |
lanl preprint quant-ph/0208195
- Brun, Carteret, et al.
- 2002
(Show Context)
Citation Context ...cuted for the circle, line and hypercube by Kendon and Tregenna [48] and extensive analytical expressions for the walk on the line with decoherence of the coin space have been obtained by Brun et al. =-=[49]-=-{. They find that the classical behaviour sets in very rapidly. As an indicator we can look at the variance of the walk. In the purely quantum case we have seen that s 2 * T 2 , whereas in the classic... |

1 |
lanl preprint quant-ph/0210180
- Brun, Carteret, et al.
- 2002
(Show Context)
Citation Context ...tor we can look at the variance of the walk. In the purely quantum case we have seen that s 2 * T 2 , whereas in the classical case s 2 * T. For the walk on the line with coin decoherence Brun et al. =-=[49, 50]-=- compute the variance as a function of the transition parameter p meas to find that even for very small values of p meas (very ‘close’ to quantum) the variance is already linear. Kendon’s numerical re... |

1 |
lanl preprint quant-ph/0210161
- Brun, Carteret, et al.
- 2002
(Show Context)
Citation Context ...evel off gradually with increasing p meas, but this does not seem to be the case. There is another interesting option to transition, the coined walk between quantum and classical taken by Brun et al. =-=[49, 51]-=-. Imagine the quantum discrete time random walk with the modification that at each iteration we use a fresh coin Hilbert space to flip our coin. This does not allow the quantum coherences between the ... |

1 | Kempe obtained her masters degrees in Algebra (1995) and Theoretical Physics (1996) in Paris, France and a PhD in Mathematics (2001) from the - Comput |