Quantum to Classical Transition for Random Walks

Brun, Todd A.; Carteret, Hilary A.; Ambainis, Andris

doi:10.1103/physrevlett.91.130602

Cited by 190 publications

(239 citation statements)

References 8 publications

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“…Only if a new coin is used for every step of the walk does it become equivalent to the classical random walk. This is in contrast with the behaviour obtained by decohering the coin (Brun et al 2003a), which always results in classical limiting behaviour, as pointed out in Brun et al (2003b). Classical behaviour is thus associated with an environment so large that one never comes close to the Poincaré recurrence time over the timescales considered.…”

Section: Multiple Coins In the Walk On The Linecontrasting

confidence: 46%

“…This can then be integrated by parts. Brun et al (2003b) chose pure dephasing for the form of the decoherence on the coin, so the coin projectors in equation (55) are…”

Section: Dephasing the Coin In The Walk On The Linementioning

confidence: 99%

“…This will also provide an example of how to use Fourier transforms to simplify the dynamics and obtain the solution. We follow the method of Brun et al (2003aBrun et al ( 2003b to solve equation (48) for the coined quantum walk on a line with dephasing applied to the coin. We thus have S given by equation (6),…”

Section: Dephasing the Coin In The Walk On The Linementioning

confidence: 99%

See 2 more Smart Citations

Decoherence in quantum walks – a review

Kendon¹

2007

Math. Struct. in Comp. Science

327

385

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The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

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Section: Multiple Coins In the Walk On The Linecontrasting

confidence: 46%

“…This can then be integrated by parts. Brun et al (2003b) chose pure dephasing for the form of the decoherence on the coin, so the coin projectors in equation (55) are…”

Section: Dephasing the Coin In The Walk On The Linementioning

confidence: 99%

See 1 more Smart Citation

Decoherence in quantum walks – a review

Kendon¹

2007

Math. Struct. in Comp. Science

327

385

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“…In other words, the opportunity to achieve exponential speedups over classical techniques by harnessing entanglement between densely encoded states in a quantum computer. Quantum walks have been the focus of several recent studies (see for example, [1,2,3,4,5]), with particular interest in possible algorithmic applications of the walks [6,7,8,9,10]. A few such algorithms have already been developed, perhaps the most notable being the 'glued trees' algorithm developed by Childs et al [6], in which quantum walks are shown to traverse a family of graphs exponentially faster than any possible classical algorithm, given a certain quantum oracle.…”

Section: Introductionmentioning

confidence: 99%

Efficient quantum circuit implementation of quantum walks

Douglas

Wang

2009

Phys. Rev. A

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Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the 'glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.

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“…The dynamics of quantum walks of both types has been studied in detail for walks on an infinite linefor the continuous-time case in Refs. [7,11,12] and for the discrete-time case in [13,14,15,16,17]. There has also been considerable work on other regular graphs.…”

Section: Introductionmentioning

confidence: 99%

Quantum walks on quotient graphs

2007

Self Cite

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A discrete-time quantum walk on a graph Γ is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. The quotient graph is constructed by removing the symmetries of the subgroup H from the original graph. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A 74, 042334 (2006)] that the hitting time for certain initial states of a quantum walks can be infinite, in contrast to classical random walks. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with hitting times which are short compared to classical random walks correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.

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Quantum to Classical Transition for Random Walks

Cited by 190 publications

References 8 publications

Decoherence in quantum walks – a review

Decoherence in quantum walks – a review

Efficient quantum circuit implementation of quantum walks

Quantum walks on quotient graphs

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