Coined quantum walks on weighted graphs

Wong, Thomas G.

doi:10.1088/1751-8121/aa8c17

Cited by 24 publications

(26 citation statements)

References 24 publications

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“…Here, C is the Grover diffusion coin [5] that inverts the amplitudes of the coin states of each vertex about their average, and S is the flip-flop shift [13] that causes the particle to hop and turn around, so S|uv = |vu . For a detailed definition of the coined quantum walk for both regular and irregular graphs, see [14].…”

Section: Introductionmentioning

confidence: 99%

Quantum walk search on the complete bipartite graph

Rhodes

Wong

2019

Phys. Rev. A

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The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of N vertices for one of k marked vertices with different initial states. We prove intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one timestep to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/2, but if they are in different partite sets, it instead reaches 1. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schrödinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.

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Section: Introductionmentioning

confidence: 99%

Quantum walk search on the complete bipartite graph

Rhodes

Wong

2019

Phys. Rev. A

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“…Since then, generalizations have been introduced that allow a randomly walking quantum particle to stay put, including lazy quantum walks [7,8] and lackadaisical quantum walks [9,10]. Here, we focus on the lackadaisical quantum walk, where a single, weighted self-loop is added to each vertex [11]. This can cause faster dispersion on the line [11][12][13], and it also speeds up quantum search on the complete graph [10,11] and two-dimensional periodic square lattice (or discrete torus) [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…Here, we focus on the lackadaisical quantum walk, where a single, weighted self-loop is added to each vertex [11]. This can cause faster dispersion on the line [11][12][13], and it also speeds up quantum search on the complete graph [10,11] and two-dimensional periodic square lattice (or discrete torus) [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Search by lackadaisical quantum walks with nonhomogeneous weights

Rhodes

Wong

2019

Phys. Rev. A

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The lackadaisical quantum walk, which is a quantum walk with a weighted self-loop at each vertex, has been shown to speed up dispersion on the line and improve spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, owing to each graph's vertex-transitivity. In this paper, we propose lackadaisical quantum walks where the self-loops have different weights. We investigate spatial search on the complete bipartite graph, which can be irregular with N1 and N2 vertices in each partite set, and this naturally leads to self-loops in each partite set having different weights l1 and l2, respectively. We analytically prove that for large N1 and N2, if the k marked vertices are confined to, say, the first partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its non-lackadaisical value when l1 = kN2/2N1 and N2 > (3 − 2 √ 2)N1, regardless of l2. When the initial state is stationary under the quantum walk, however, then the success probability is improved when l1 = kN2/2N1, now without a constraint on the ratio of N1 and N2, and again independent of l2. Next, when marked vertices lie in both partite sets, then for either initial state, there are many configurations for which the self-loops yield no improvement in quantum search, no matter what weights they take.

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“…Since the seminal paper by Shenvi et al [22], coined quantum walks are being used for search algorithms [23]. Search on the two-dimensional square lattice was analyzed in [9,24,25]. Search on the hexagonal lattice was analyzed in [20], which showed that the optimal number of steps is O( √ N ln N ) and the success probability is O(1/ ln N ), where N is the number of sites, and quantum transport on the hexagonal lattice was analyzed in [26].…”

Section: Introductionmentioning

confidence: 99%

Staggered quantum walk on hexagonal lattices

et al. 2018

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A discrete-time staggered quantum walk was recently introduced as a generalization that allows to unify other versions, such as the coined and Szegedy's walk. However, it also produces new forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.

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Coined quantum walks on weighted graphs

Cited by 24 publications

References 24 publications

Quantum walk search on the complete bipartite graph

Quantum walk search on the complete bipartite graph

Search by lackadaisical quantum walks with nonhomogeneous weights

Staggered quantum walk on hexagonal lattices

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