Full revivals in 2D quantum walks

Štefaňák, Martin; Kollár, B.; Kiss, T.; Jex, Igor

doi:10.1088/0031-8949/2010/t140/014035

Cited by 25 publications

(20 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, the trapping in DTQWs is caused by the presence of flat bands in the quasi-energy spectrum of the walk [22], which naturally depends on the choice of the coin operator. It was shown that the spectra of all trapping two-dimensional DTQWs contain at least two such flat bands with a π difference between their quasienergies [24]. Finding trapping coins involves solving Eq.…”

mentioning

confidence: 99%

Strongly trapped two-dimensional quantum walks

2015

View full text Add to dashboard Cite

Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a broad scope of applications. In particular, they can be used as computational primitives, and they are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due to their quantumness. In some cases, however, they can remain localized at their initial state (trapping). The trapping and other fundamental properties of DTQWs are determined by the choice of the coin operator. We introduce and analyze an up to now uncharted type of walks driven by a coin class leading to strong trapping, complementing the known list of walks. This class of walks exhibit a number of exciting properties with the possible applications ranging from light pulse trapping in a medium to topological effects and quantum search.Comment: 5 pages, 4 figures, Accepted for publication in Physical Review

show abstract

mentioning

confidence: 99%

Strongly trapped two-dimensional quantum walks

2015

View full text Add to dashboard Cite

show abstract

“…in Ref. [17], [18], [1] and others. In this case also the distribution of the probability of finding the particle spreads away from the centre in general.…”

Section: Introductionmentioning

confidence: 86%

An effective Hamiltonian approach to quantum random walk

et al. 2017

View full text Add to dashboard Cite

In this article we present an effective Hamiltonian approach for Discrete Time Quantum Random Walk. A form of the Hamiltonian for one dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are the generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative like that of Ref. [1]. We showed that in case of two-step walk, effectively the time evolution operator can have multiplicative form. In case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the Graphene Hamiltonian the walk has been studied on a Graphene lattice and we conclude the preference of additive approach over the multiplicative one.

show abstract

“…Any part of the initial state that has an overlap with these states will not participate in the otherwise ballistic spreading. The existence of the trapped states is due to the following pairs of eigenstates of the unitary evolution operator [13] |ϕ ± x,y =…”

Section: Applicationsmentioning

confidence: 99%

“…(41) for the double line quantum walk. This saves a lot of effort in comparison with re-deriving these results from the corresponding definitions using techniques similar to [13].…”

Section: Applicationsmentioning

confidence: 99%

Projection Theorem for Discrete-Time Quantum Walks

Potoček

2020

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for a generality of the presented statements. Using this approach we also identify some pathological cases in which the projection mapping breaks down.For walks on lattices, the operation typically results in quantum walks with hyper-dimensional coin spaces. Such walks can, conversely, be viewed as projections of walks on inaccessible, larger spaces, and their properties can be inferred from the parental walk. We show that this is is the case for a lazy quantum walk, a walk with large coherent jumps and a walk on a circle with a twisted boundary condition. We also discuss the relation of this theory to the time-multiplexing optical implementations of quantum walks. Moreover, this manifestly irreversible operation can, in some cases and with a minor adjustment, be undone, and a quantum walk can be reconstructed from a set of its projections.

show abstract

Full revivals in 2D quantum walks

Cited by 25 publications

References 22 publications

Strongly trapped two-dimensional quantum walks

Strongly trapped two-dimensional quantum walks

An effective Hamiltonian approach to quantum random walk

Projection Theorem for Discrete-Time Quantum Walks

Contact Info

Product

Resources

About