One-dimensional quantum walks via generating function and the CGMV method

Konno, Norio; Segawa, Etsuo

doi:10.26421/qic14.13-14-8

Cited by 11 publications

(9 citation statements)

References 18 publications

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“…In recent years, quantum walks have been studied extensively; see [1,11,14,15,16,17,18,19,21,28,29,30,32,33,34,42] for some papers on this subject that have appeared in recent years. In the case of one-dimensional coined quantum walks, a substantial amount of progress has been made by using the beautiful observation of Cantero, Grünbaum, Moral, and Velázquez which relates the unitary time-one maps of such quantum walks with CMV matrices, a class of unitary operators which arise from a separate construction in the theory of orthogonal polynomials on the unit circle (OPUC) [14].…”

Section: Introductionmentioning

confidence: 99%

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

Fillman

Ong

Zhang

2016

Commun. Math. Phys.

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We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.

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Section: Introductionmentioning

confidence: 99%

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

Fillman

Ong

Zhang

2016

Commun. Math. Phys.

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“…CMV walks not only constitute simple quantum walk models, but they are universal models for quantum walks, since any unitary operator-as the unitary step governing the evolution of a quantum walk-has a CMV representation. This paves the way to the use of the CMV representation as a resource for the analysis of general quantum walks, a technique nowadays known as the "CGMV method" [CGMV12,Kon11,KS14].…”

Section: Introductionmentioning

confidence: 99%

Occupation time for classical and quantum walks

Grunbaum¹,

Velazquez²,

Wilkening³

2020

Preprint

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This is a personal tribute to Lance Littlejohn on the occasion of his 70th birthday. It is meant as a present to him for many years of friendship. It is not written in the "Satz-Beweis" style of Edmund Landau or even in the format of a standard mathematics paper. It is rather an invitation to a fairly new, largely unexplored, topic in the hope that Lance will read it some afternoon and enjoy it.If he cares about complete proofs he will have to wait a bit longer; we almost have them but not in time for this volume. We hope that the figures will convince him and other readers that the phenomena displayed here are interesting enough.

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“…In recent years, quantum walks have been studied extensively; see [1,2,5,6,7,11,16,26,29,30,31,33,34,35,42,46] for some papers on this subject that have appeared in the past five years. These are quantum analogues of classical random walks.…”

Section: Introductionmentioning

confidence: 99%

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Damanik

Fillman

Ong

2016

Journal de Mathématiques Pures et Appliquées

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We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero-Grünbaum-Moral-Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowledge. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.

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One-dimensional quantum walks via generating function and the CGMV method

Cited by 11 publications

References 18 publications

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

Occupation time for classical and quantum walks

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

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