One-dimensional coinless quantum walks

Portugal, Renato; Boettcher, Stefan; Falkner, Stefan

doi:10.1103/physreva.91.052319

Cited by 30 publications

(32 citation statements)

References 25 publications

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“…with coin C and shift S. Unitarity, U † U = I, demands [34,35] that the coin be a unitary matrix of rank r > 1, such that the site amplitudes ψ x,t become complex r-dimensional vectors in "coin" space. For simplicity, this quantum walk is commonly studied on networks of regular degree r for all x, so that the same coin can be applied at every site.…”

Section: Discrete-time Quantum Walksmentioning

confidence: 99%

Relation between random walks and quantum walks

2015

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Based on studies of four specific networks, we conjecture a general relation between the walk dimensions d w of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that d w of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution ρ(x,t) in space and time of this quantum-walk equation exhibits a scaling collapse for a variable x dw /t in the weak limit, which defines d w and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for ρ (x,t) in each case with extensive numerical simulation. The exact values for d w themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.

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Section: Discrete-time Quantum Walksmentioning

confidence: 99%

Relation between random walks and quantum walks

2015

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“…The evolution operator analyzed in Ref. [19] is obtained if we take α = 0 and θ = π/2 and the one analyzed in Ref. [21] is obtained if we take α = 0.…”

Section: Oriented Graphsmentioning

confidence: 92%

Discrete-Time Quantum Walks on Oriented Graphs

Chagas

Portugal²

2020

Electron. Proc. Theor. Comput. Sci.

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The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called α, that quantifies the amount of orientation. We show that the parameter α can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.

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“…In this manuscript, we explore the second way, starting with the minimal coin dimension s = 1, and performing a systematic analysis of the Euclidean scenario. QWs with a one-dimensional coin are often referred to as scalar or coinless [38,[43][44][45][46]. Despite the algorithmic simplicity of the model, finding all scalar QWs for an arbitrary graph is not a straightforward task.…”

Section: Introductionmentioning

confidence: 99%

Chirality from quantum walks without a quantum coin

2019

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Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom-called the coin and represented by a finite-dimensional Hilbert space-is associated to each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.

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One-dimensional coinless quantum walks

Cited by 30 publications

References 25 publications

Relation between random walks and quantum walks

Relation between random walks and quantum walks

Discrete-Time Quantum Walks on Oriented Graphs

Chirality from quantum walks without a quantum coin

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