## Decoherence in discrete quantum walks (2003)

Venue: | Selected Lectures from DICE 2002. Lecture Notes in Physics, 633:253–267 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Kendon03decoherencein,

author = {Viv Kendon and Ben Tregenna},

title = {Decoherence in discrete quantum walks},

booktitle = {Selected Lectures from DICE 2002. Lecture Notes in Physics, 633:253–267},

year = {2003},

pages = {030118--2}

}

### OpenURL

### Abstract

Abstract. We present an introduction to coined quantum walks on regular graphs, which have been developed in the past few years as an alternative to quantum Fourier transforms for underpinning algorithms for quantum computation. We then describe our results on the effects of decoherence on these quantum walks on a line, cycle and hypercube. We find high sensitivity to decoherence, increasing with the number of steps in the walk, as the particle is becoming more delocalised with each step. However, the effect of a small amount of decoherence can be to enhance the properties of the quantum walk that are desirable for the development of quantum algorithms, such as fast mixing times to uniform distributions. 1 Introduction to Quantum Walks Quantum walks are based on a generalisation of classical random walks, which have found many applications in the field of computing. Examples of the power of classical random walks to solve hard problems include algorithms for solving k-SAT [1], estimating the volume of a convex body [2], and approximation of

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Citation Context ...quivalence to quantum Turing machines has been shown [4]. Most known quantum algorithms are based on the quantum Fourier transform, for an introduction to quantum computing and algorithms see, e. g., =-=[5, 6]-=-. Quantum versions of random walks provide a distinctly different paradigm in which to develop quantum algorithms. Very recently, two such algorithms have been presented. Shenvi et al. [7] proved that... |

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Citation Context ...puting. Examples of the power of classical random walks to solve hard problems include algorithms for solving k-SAT [1], estimating the volume of a convex body [2], and approximation of the permanent =-=[3]-=-. They are a subset of a wider model of computation, cellular automata, which have been proved universal for classical computation. The utility of classical walks suggests that extending the formalism... |

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Citation Context ...on of classical random walks, which have found many applications in the field of computing. Examples of the power of classical random walks to solve hard problems include algorithms for solving k-SAT =-=[1]-=-, estimating the volume of a convex body [2], and approximation of the permanent [3]. They are a subset of a wider model of computation, cellular automata, which have been proved universal for classic... |

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Citation Context ...nd many applications in the field of computing. Examples of the power of classical random walks to solve hard problems include algorithms for solving k-SAT [1], estimating the volume of a convex body =-=[2]-=-, and approximation of the permanent [3]. They are a subset of a wider model of computation, cellular automata, which have been proved universal for classical computation. The utility of classical wal... |

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Citation Context ...ntly, two such algorithms have been presented. Shenvi et al. [7] proved that a quantum walk can perform the same task as Grover’s search algorithm [8], with the same quadratic speed up. Childs et al. =-=[9]-=- describe a quantum algorithm for transversing a particular graph exponentially faster than can be done classically. This exponential speed up is very promising, though the problem presented is somewh... |

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Citation Context ... Farhi and Gutmann [10], is non-local, in the sense that there is a small probability of the particle moving arbitrarily far away in a given unit time interval. The quantum cellular automata of Meyer =-=[15, 16]-=- are not homogeneous, in that the full dynamics is specified over two time steps rather than one. One may make the quantum walk non-unitary by measuring the particle at each step, but this simply repr... |

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Citation Context ...at contrived. In fact, several possible extensions of classical random walks to the quantum regime have been proposed [10–12], however, here we will only treat the discrete time, coined quantum walks =-=[13]-=-, subsequently these are referred to simply as “quantum walks”. Before introducing quantum walks, it is helpful to review the properties of classical random walks. This is followed by an overview of q... |

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Citation Context ..., e. g., [5, 6]. Quantum versions of random walks provide a distinctly different paradigm in which to develop quantum algorithms. Very recently, two such algorithms have been presented. Shenvi et al. =-=[7]-=- proved that a quantum walk can perform the same task as Grover’s search algorithm [8], with the same quadratic speed up. Childs et al. [9] describe a quantum algorithm for transversing a particular g... |

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Citation Context ...we note that there is an equivalent formulation of the coined quantum walk in terms of a simple quantum process on a directed graph, derived from the original undirected graph, that is due to Watrous =-=[17]-=-. 1.3 Coined Quantum Walks By analogy with the classical walk, in which one pictures flipping a coin at each time-step to determine which edge to leave the current vertex by, interesting quantum resul... |

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Citation Context ...quivalence to quantum Turing machines has been shown [4]. Most known quantum algorithms are based on the quantum Fourier transform, for an introduction to quantum computing and algorithms see, e. g., =-=[5, 6]-=-. Quantum versions of random walks provide a distinctly different paradigm in which to develop quantum algorithms. Very recently, two such algorithms have been presented. Shenvi et al. [7] proved that... |

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Citation Context ...arkovian) The quantum transition matrix U, equivalent to M, should have similar properties to M. In addition, it should be • unitary because all pure quantum processes are unitary. As proved by Meyer =-=[14, 15]-=-, this requirement of unitarity is incompatible with the three prior properties, except in very special circumstances that don’t produce interesting quantum dynamics. In order to generate a non-trivia... |

12 |
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Citation Context ... of this paper has been presented more fully in [25]. i 2.1 Decoherence in a Quantum Walk on a Line Since there is an experimental proposal to implement a quantum walk on a line in an optical lattice =-=[27]-=-, as well as the three examples for IPi given above, we considered the likely form of experimental errors, and also modeled the effect of an imperfect Hadamard on the coin. The Hadamard operation may ... |

11 |
Decoherence can be useful in quantum walks
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(Show Context)
Citation Context ... . (17) Bounds have also been established in the case of more general graphs and it is conjectured that mixing times can be improved at most polynomially by quantum walks [13]. Numerical studies, see =-=[25]-=-, suggest the form of the quantum mixing time is actually MQ ε = O (N/ε), and explain why tighter analytical bounds are hard to obtain. 1.6 Coined Quantum Walk on a Hypercube A hypercube of dimension ... |

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Citation Context ... deviation σQ is thus linear in T, in contrast to √ T for the classical walk. Quantum walks on a line have now been studied in considerable detail. Discussions of absorbing boundaries have been given =-=[19, 20, 22]-=-, with applications to halting problems in mind. Extensions to multiple coins have been made by Brun et al. [23, 24]. However, though useful for understanding the basic properties of quantum walks, th... |

8 | Unitarity in one dimensional nonlinear quantum cellular automata. quant-ph/9605023
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(Show Context)
Citation Context ...arkovian) The quantum transition matrix U, equivalent to M, should have similar properties to M. In addition, it should be • unitary because all pure quantum processes are unitary. As proved by Meyer =-=[14, 15]-=-, this requirement of unitarity is incompatible with the three prior properties, except in very special circumstances that don’t produce interesting quantum dynamics. In order to generate a non-trivia... |

8 |
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Citation Context ...he most natural choice given the symmetries of the hypercube. Instead, a biased coin has been selected, which distinguishes the edge along which the particle arrived at the vertex from all the others =-=[26]-=-. This “Grover” coin acts thus, G|a〉 = 2 N ∑ |b〉 − |a〉 . (19) The full evolution for a single time step of this walk is then given by U = T · (G ⊗ I). The quantum walk on the hypercube can be solved a... |

8 | Tregenna, Decoherence in a quantum walk on the line
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Citation Context ...ulated σp(T) analytically for pT ≪ 1 and T ≫ 1 for the case where IPi is the projector onto the preferred basis {|a, x〉}, i. e., the decoherence affecting both particle and coin. Details are given in =-=[28]-=-), the result is [ σp(T) ≤ σ(T) 1 − pT 6 √ 2 + p √ (1 − 1/ 2 √ 2) + O(p 2 ] , 1/T) . (22) The first order dependence is thus proportional to pT, so the sensitivity to decoherence grows linearly in T f... |

3 |
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Citation Context ...e interesting quantum dynamics. In order to generate a non-trivial quantum evolution, one or more of these constraints must be relaxed. The continuous time quantum walks proposed by Farhi and Gutmann =-=[10]-=-, is non-local, in the sense that there is a small probability of the particle moving arbitrarily far away in a given unit time interval. The quantum cellular automata of Meyer [15, 16] are not homoge... |

2 |
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Citation Context ...uantum Walk on a N-Cycle We now consider a walk on a N-cycle subjected to decoherence. There is an experimental proposal for implementation of a quantum walk on a cycle in the phase of a cavity field =-=[29]-=-, in which further aspects of decoherence in such quantum walks are considered. Recall from Sect. 1.5 that the pure quantum walk on a cycle with N odd, is known [13] to mix in time ≤ O(N log N) if a H... |

1 | Grossing and A Zeilinger. Quantum cellular automata - unknown authors - 1988 |

1 |
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Citation Context ...cases standard techniques from classical graph theory have been applied with some success, namely solution in the Fourier space of the problem [13, 19], and also the technique of generating functions =-=[20]-=-. Fourier solutions are possible when the graph is of a particular form, known as a Cayley graph. Any discrete group has an associated Cayley graph, in which the elements of the group form the vertice... |

1 |
H A Carteret, and Andris Ambainis. The quantum to classical transition for random
- Brun
- 2002
(Show Context)
Citation Context ...ied in considerable detail. Discussions of absorbing boundaries have been given [19, 20, 22], with applications to halting problems in mind. Extensions to multiple coins have been made by Brun et al. =-=[23, 24]-=-. However, though useful for understanding the basic properties of quantum walks, the walk on a line is too simple to yield interesting quantum problems for significant algorithms. 1.5 Coined Quantum ... |

1 |
H A Carteret, and Andris Ambainis. Quantum random walks with decoherent coins. quant-ph/0210180
- Brun
- 2002
(Show Context)
Citation Context ...ied in considerable detail. Discussions of absorbing boundaries have been given [19, 20, 22], with applications to halting problems in mind. Extensions to multiple coins have been made by Brun et al. =-=[23, 24]-=-. However, though useful for understanding the basic properties of quantum walks, the walk on a line is too simple to yield interesting quantum problems for significant algorithms. 1.5 Coined Quantum ... |

1 |
Tuning quantum walks: coins and initial states
- Tregenna, Flanagan, et al.
- 2003
(Show Context)
Citation Context ...if a Hadamard coin is used. The quantum walk on a cycle with N even does not mix to the uniform distribution with a Hadamard coin, but can be made to do so by appropriate choice of coin flip operator =-=[30]-=-. Under the action of a small amount of decoherence, the mixing time becomes shorter for all cases, typical results are shown in Fig. 5. In particular, decoherence causes the even-N cycle to mix to th... |