Understanding and controlling<i>N</i>-dimensional quantum walks via dispersion relations: application to the two-dimensional and three-dimensional Grover walks—diabolical points and more

Hinarejos, M.; Perez, A.; Roldán, Eugenio; Romanelli, A.; Valcárcel, Germán J. de

doi:10.1088/1367-2630/15/7/073041

Cited by 10 publications

(21 citation statements)

References 74 publications

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“…The analysis that was used to obtain an approximate expression for the functions I i (α, t) starts with the expansion Eq. (45), that only depends on the dispersion relation. Thus, these functions are the same for the QW and the DQCA, once the mapping (18) is established.…”

Section: B Stationary Phase Methodsmentioning

confidence: 99%

Asymptotic properties of the Dirac quantum cellular automaton

Perez

2016

Phys. Rev. A

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We show that the Dirac quantum cellular automaton [Ann. Phys. 354 (2015) 244] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter, which plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions, and are similar to the ones achieved by highly localized states of the Dirac equation.Comment: 15 pages, 6 figure

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Section: B Stationary Phase Methodsmentioning

confidence: 99%

Asymptotic properties of the Dirac quantum cellular automaton

Perez

2016

Phys. Rev. A

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“…Calculating analytically the evolution of a quantum walker over many steps is generally a demanding task. However, for translationally invariant systems, it is possible to characterize the long time asymptotic dynamics in a simple way by analyzing the dispersion relation, i.e., the k-dependent quasienergies ω(k) of the unitary evolution operator obtained after performing the spatial Fourier transform [60,61]. By locating all local extrema of the group velocities defined as the derivative v g (k) = dω(k)/dk we can determine the number and propagation speeds of wavefronts emerging from an initially localized state.…”

Section: Dynamical Features Of Four-dimensional Coinsmentioning

confidence: 99%

Photonic quantum walks with four-dimensional coins

Lorz

Meyer-Scott

Nitsche

et al. 2019

Phys. Rev. Research

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The dimensionality of the internal coin space of discrete-time quantum walks has a strong impact on the complexity and richness of the dynamics of quantum walkers. While two-dimensional coin operators are sufficient to define a certain range of dynamics on complex graphs, higher-dimensional coins are necessary to unleash the full potential of discrete-time quantum walks. In this work, we present an experimental realization of a discrete-time quantum walk on a line graph that, instead of two-dimensional, exhibits a four-dimensional coin space. Making use of the extra degree of freedom we observe multiple ballistic propagation speeds specific to higher-dimensional coin operators. By implementing a scalable technique, we demonstrate quantum walks on circles of various sizes, as well as on an example of a Husimi cactus graph. The quantum walks are realized via time-multiplexing in a Michelson interferometer loop architecture, employing as the coin degrees of freedom the polarization and the traveling direction of the pulses in the loop. Our theoretical analysis shows that the platform supports implementations of quantum walks with arbitrary 4 × 4 unitary coin operations, and usual quantum walks on a line with various periodic and twisted boundary conditions.

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“…(22), that is designed to be a separable matrix, but if the same matrix is applied to other ordering, e.g. ours in [7], then one is not implementing a separable operator, and one is indeed making a different QW. In fact, if one is willing to implement the Hadamard walk with the ordering of the coin elements in [7], the correct matrix must be written aŝ…”

Section: Appendix Bmentioning

confidence: 99%

“…The quantum walk (QW) [1][2][3][4] is a simple quantum diffusion model that can be understood as the quantum analog of classical random walks [5], but also as the discretized version of different wave equations including the Schrödinger [6,7] and the Dirac [8][9][10] equations. Appearing in two basic forms, namely the coined QW [11,12] and the continuous QW [13,14], here we deal with the former.…”

Section: Introductionmentioning

confidence: 99%

Electric quantum walks in two dimensions

et al. 2016

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We study electric quantum walks in two dimensions considering Grover, Alternate, Hadamard, and DFT quantum walks. In the Grover walk the behaviour under an electric field is easy to summarize: when the field direction coincides with the x or y axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect 2D trapping is frustrated. The analysis of the alternate walk helps to understand the behaviour of the Grover walk as both walks are partially equivalent; in particular, it helps to understand the role played by the existence of conical intersections in the dispersion relations, as we show that when these are removed a perfect 2D trapping can occur for suitable directions of the field. We complete our study with the electric DFT and Hadamard walks in 2D, showing that the latter can exhibit perfect 2D trapping.Comment: References added and minor typos correcte

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Understanding and controllingN-dimensional quantum walks via dispersion relations: application to the two-dimensional and three-dimensional Grover walks—diabolical points and more

Cited by 10 publications

References 74 publications

Asymptotic properties of the Dirac quantum cellular automaton

Asymptotic properties of the Dirac quantum cellular automaton

Photonic quantum walks with four-dimensional coins

Electric quantum walks in two dimensions

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