Spatial Search on Johnson Graphs by Continuous-Time Quantum Walk

Tanaka, Hajime; Sabri, Mohamed; Portugal, Renato

doi:10.48550/arxiv.2108.01992

Cited by 3 publications

(6 citation statements)

References 19 publications

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“…For later use, we first recall the subspace of the Hilbert space H V = span v : v ∈ V used in the spatial search by continuous-time QW [18]. Consider the following subsets of V:…”

Section: The Spatial Search Algorithmmentioning

confidence: 99%

“…As in [18], we invoke the implicit function theorem for complex analytic functions [23]. Extend for the moment the range of ǫ to complex numbers with |ǫ| 2 < (2k − 1) −1 , so that the functions p ℓ (ǫ) and e ± ℓ (ǫ) are all analytic.…”

Section: An Eigenbasis Of the Invariant Subspacementioning

confidence: 99%

“…Unlike in[18], we do not normalize these vectors here 2. The numbers a ℓ , b ℓ , and c ℓ are known as the intersection numbers of J(n, k); see[19,21,22].…”

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confidence: 99%

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Spatial Search on Johnson Graphs by Discrete-Time Quantum Walk

Tanaka,

Sabri,

Portugal

2021

Preprint

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The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) The walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker's position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex-and distancetransitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π √ N /(2 √ 2) steps, where N is the number of vertices of the Johnson graph.

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“…For later use, we first recall the subspace of the Hilbert space H V = span v : v ∈ V used in the spatial search by continuous-time QW [18]. Consider the following subsets of V:…”

Section: The Spatial Search Algorithmmentioning

confidence: 99%

Section: An Eigenbasis Of the Invariant Subspacementioning

confidence: 99%

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Spatial Search on Johnson Graphs by Discrete-Time Quantum Walk

Tanaka,

Sabri,

Portugal

2021

Preprint

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“…For distance regular graphs with larger diameter, Wong [17] and then Tanaka et al [15] proved that the Johnson graphs J(n, k), for constant k ≥ 3, have spatial search. Since the constant spectral gap condition holds for Johnson graphs, this provides an alternative and immediate proof that they have optimal spatial search.…”

Section: Introductionmentioning

confidence: 99%

Of Shadows and Gaps in Spatial Search

Chan¹,

Godsil²,

Tamon³

et al. 2022

Preprint

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Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty et al. (Physical Review A, 102:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

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“…While this idea was formalized by Shenvi, Kempe and Whaley [14], its first application dates back to 1996, when Grover [11] showed that finding a marked vertex in a looped K n takes O( √ n) steps of a quantum walk. Since then, quantum walk search has been studied on various graphs, including hypercubes [14], Cartesian powers of cycles [6], strongly regular graphs [13], certain Johnson graphs [2,19,20,16], and more generally, regular locally arc-transitive graphs [12]. For most of these graphs, quantum walks arrive at the marked vertices faster than the classical random walks.…”

Section: Introductionmentioning

confidence: 99%

The average search probabilities of discrete-time quantum walks

Zhan¹

2021

Preprint

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We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertexdeleted subgraph. In particular, for any family of • complete graphs, or• strongly regular graphs, or• distance regular graphs of a fixed parameter d, varying valency k and varying size n, such that k d−1 /n vanishes as k increases, the average search probability approaches 1/4 as the valency goes to infinity. We also present a more relaxed criterion, in terms of the intersection array, for this limit to be approached by distance regular graphs.

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Spatial Search on Johnson Graphs by Continuous-Time Quantum Walk

Cited by 3 publications

References 19 publications

Spatial Search on Johnson Graphs by Discrete-Time Quantum Walk

Spatial Search on Johnson Graphs by Discrete-Time Quantum Walk

Of Shadows and Gaps in Spatial Search

The average search probabilities of discrete-time quantum walks

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