Symmetry of Distribution for the One-Dimensional Hadamard Walk

Konno, Norio; Namiki, Takao; Soshi, Takahiro

doi:10.4036/iis.2004.11

Cited by 32 publications

(44 citation statements)

References 28 publications

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“…This choice results in a symmetric evolution with P n (t) = P −n (t) [22]. The position and chirality of the walker are jointly measured every T steps.…”

Section: Periodic Measurementsmentioning

confidence: 99%

Decoherence in the quantum walk on the line

Romanelli

Siri

Abal

et al. 2005

Physica A: Statistical Mechanics and its Applications

128

141

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We investigate the quantum walk on the line when decoherences are introduced either through simultaneous measurements of the chirality and particle position, or as a result of broken links. Both mechanisms drive the system to a classical diffusive behavior. In the case of measurements, we show that the diffusion coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement. When links between neighboring sites are randomly broken with probability p per unit time, the evolution becomes decoherent after a characteristic time that scales as 1/p. The fact that the quadratic increase of the variance is eventually lost even for very small frequencies of disrupting events, suggests that the implementation of a quantum walk on a real physical system may be severely limited by thermal noise and lattice imperfections.

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“…This choice results in a symmetric evolution with P n (t) = P −n (t) [22]. The position and chirality of the walker are jointly measured every T steps.…”

Section: Periodic Measurementsmentioning

confidence: 99%

Decoherence in the quantum walk on the line

Romanelli

Siri

Abal

et al. 2005

Physica A: Statistical Mechanics and its Applications

128

141

View full text Add to dashboard Cite

show abstract

“…In much the same way as we now know almost everything about the properties and possible states of two qubits -though quantum computers will clearly need far more than two qubits to be useful -the simple quantum walk on a line has now been thoroughly studied (see, for example, Ambainis et al (2001), Bach et al (2004), Yamasaki et al (2002), Kendon and Tregenna (2003), Brun et al (2003a;2003c), Konno et al (2004), Konno (2002) and Carteret et al (2003)), though there is no suggestion that it will lead to useful quantum walk algorithms by itself.…”

Section: Coined Quantum Walk On An Infinite Linementioning

confidence: 99%

“…It is also asymmetric, with the asymmetry determined by the initial coin state. A symmetric distribution can be obtained by choosing the initial coin state as either (|−1 ± i|+1 )/ √ 2, or cos(π/8)|−1 − sin(π/8)|+1 , see Konno et al (2004) and Tregenna et al (2003). The moments have been calculated in Ambainis et al (2001): for asymptotically large times T for a walk starting at the origin,…”

Section: Coined Quantum Walk On An Infinite Linementioning

confidence: 99%

Decoherence in quantum walks – a review

Kendon¹

2007

Math. Struct. in Comp. Science

323

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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“…The numerical simulations by these two groups suggest that the probability distribution of the disordered QW converges to a binomial distribution by averaging over many trials. So the main purpose of the letter is to prove the above numerical results by using a path integral approach, which has been used in [6,7,8], for example. In fact, our theorem (see Theorem 1) shows that the expectation of the probability distribution for the disordered QW becomes the probability distribution of a classical symmetric random walk.…”

Section: Introductionmentioning

confidence: 99%