Localization of multi-state quantum walk in one dimension

A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely-studied coined walk. Coined walks require the direct product of the site basis with the coin space, coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.

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“…II, the former leads to evolution equations such as in (16), while the latter leads to Eq. (17). Note that in Eq.…”

Section: Generalized Quantum Walksmentioning

confidence: 99%

One-dimensional coinless quantum walks

2015

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“…Almost all classes of the DTQW except for a one-dimensional one with a two-dimensional coin [24,38,39] have the aspect of the localization defined that the limit distribution of the DTQW divided by some power of the time variable has the probability density given by the Dirac delta function. In the one dimensional DTQW, the localization was shown in the models with a three-dimensional coin [29], a four-dimensional coin [28], a two-dimensional coin with memory [56], a two-dimensional coin in a random environment [31], two-dimensional coins [48], an spatially incommensurate coin [70] and a time-dependent two-dimensional coin on the Fibonacci quantum walk [63] and a random coin [30,58]. On the other hand, the nature of the localization in the two-dimensional DTQW has been studied numerically [51,78] and analytically [27,81].…”

Section: Review Of Discrete Time Quantum Walkmentioning

confidence: 99%

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Shikano¹

2013

Jnl of Comp & Theo Nano

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The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.

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“…After that, more refined simulations were performed by Tregenna et al [15] and an exact proof on the localization was given by Inui et al [16]. The second is found in the four-state quantum walk [17,18]. In this walk, a particle moves not only the nearest sites but also the second nearest sites according to the four inner states.…”

Section: Introductionmentioning

confidence: 98%

One-dimensional three-state quantum walk

2005

Self Cite

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We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wavefunction of the particle starting from the origin for any initial qubit state, and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after long time with high probability.

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Localization of multi-state quantum walk in one dimension

Cited by 80 publications

References 13 publications

One-dimensional coinless quantum walks

One-dimensional coinless quantum walks

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

One-dimensional three-state quantum walk

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