Quantum random walk on the line as a Markovian process

Romanelli, A.; Schifino, A. C. Sicardi; Siri, R.; Abal, G.; Auyuanet, A.; Donangelo, R.

doi:10.1016/j.physa.2004.02.061

Cited by 75 publications

(97 citation statements)

References 19 publications

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“…It would be interesting to investigate the role of decoherence in continuous descriptions of the QW, such as that we have presented here. In this sense, the recent work by Romanelli et al [21], in which Markovian and intereference terms of the quantum evolution are separated, provides a new interesting tool for this type of analysis.…”

Section: Discussionmentioning

confidence: 99%

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Propagating quantum walks: the origin of interference structures

Knight¹,

Roldán²,

Sipe³

2004

Journal of Modern Optics

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We analyze the solution of the coined quantum walk on a line. First, we derive the full solution, for arbitrary unitary transformations, by using a new approach based on the four "walk fields" which we show determine the dynamics. The particular way of deriving the solution allows a rigorous derivation of a long wavelength approximation. This long wavelength approximation is useful as it provides an approximate analytical expression that captures the basics of the quantum walk and allows us to gain insight into the physics of the process. IntroductionConsider a two-state particle, e.g. a two-level atom, moving along a onedimensional lattice. The displacement is conditional; that is, the particle jumps one site at each time to the right or to the left depending on its internal state. The movement of such a particle is rather trivial: If the particle is in its "up" or "down" internal state it moves to the right or to the left, respectively, and it undergoes a superposition of right and left displacements if it is in a superposition of its "up" and "down" states. But the movement becomes nontrivial if *

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Section: Discussionmentioning

confidence: 99%

“…Because of the range of k and ω appearing in Eq. (24), we need only consider such solutions over the ranges (21). There are then two such solutions.…”

Section: Dispersion Relationmentioning

confidence: 99%

Propagating quantum walks: the origin of interference structures

Knight¹,

Roldán²,

Sipe³

2004

Journal of Modern Optics

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“…[33,34,37] for the GCD dynamics, but there a handmade technique [39,40] was used to separate the Markovian evolution from the interference term. It can also be obtained [34] from Eqs.…”

Section: Decoherence In the Qwmentioning

confidence: 99%

Spatially Dependent Decoherence and Anomalous Diffussion of Quantum Walks

Cañellas¹,

Romanelli²

2013

Jnl of Comp & Theo Nano

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We analyze the long time behavior of a discrete time quantum walk subject to decoherence with a strong spatial dependence, acting on one half of the lattice. We show that, except for limiting cases on the decoherence parameter, the quantum walk at late times behaves sub-ballistically, meaning that the characteristic features of the quantum walk are not completely spoiled. Contrarily to expectations, the asymptotic behavior is non Markovian, and depends on the amount of decoherence. This feature can be clearly shown on the long time value of the Generalized Chiral Distribution (GCD).

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“…to lowest order in q ≡ 1 − p. A simpler model of decoherence in both position and coin space by applying measurements at regular intervals and projecting the coin into the σ y basis to preserve the symmetry of the walk was solved in Romanelli et al (2004). They obtain a spreading rate that is basically classical, with a prefactor controlled by the rate of the measurements.…”

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confidence: 99%

“…They also generalise to random intervals between measurements, drawing on an analogy with Brownian motion. Romanelli et al (2004) also presents a unitary decoherence model affecting the position by analysing a quantum walk in which links between the positions on the line are broken with probability p. The transition that should have taken place is turned into a self loop for that step of the walk. The unitary operator for each step is modified to take account of the links that happen to be broken at that time step.…”

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confidence: 99%

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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Quantum random walk on the line as a Markovian process

Cited by 75 publications

References 19 publications

Propagating quantum walks: the origin of interference structures

Propagating quantum walks: the origin of interference structures

Spatially Dependent Decoherence and Anomalous Diffussion of Quantum Walks

Decoherence in quantum walks – a review

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