Adjacent vertices can be hard to find by quantum walks

Nahimovs, Nikolajs; Santos, Raqueline A. M.

doi:10.48550/arxiv.1605.05598

Cited by 3 publications

(14 citation statements)

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“…1. This figure reveals three properties, identified by Nahimovs and Rivosh [10] and further explored in follow-up work by Nahimovs and Santos [11], of stationary states. First, the directional amplitudes of unmarked vertices (depicted by single circles) are equal.…”

Section: Introductionsupporting

confidence: 60%

“…In this paper, however, we show that the stationary states described by Nahimovs, Rivosh, and Santos [10,11] are not the only stationary states. We give the exact necessary and sufficient conditions on the amplitudes between adjacent vertices for a state to be stationary.…”

Section: Introductioncontrasting

confidence: 51%

“…We do this by incorporating concepts from [8] regarding uniform and flip states, which form an orthogonal basis for directional states. Then we show that the type of stationary state described by Nahimovs, Rivosh, and Santos [10,11] is optimal, in the sense that it is the stationary states closest to the initial uniform state (1). Then we prove two theorems about the existence of stationary states on general graphs and with a connected component of marked vertices.…”

Section: Introductionmentioning

confidence: 89%

“…For example, vertices 6 and 7 point to each other with amplitude −3a. Nahimovs, Rivosh, and Santos [10,11] showed that a state with these three properties is a stationary state (i.e., 1-eigenvector) of the quantum walk search operator U (2). Going through each operator in U , the query Q flips the sign of the marked vertices, the coin C again flips the sign of the marked vertices since their average amplitudes are zero, and the shift S swaps pairs of amplitudes pointing to each other, which are equal.…”

Section: Introductionmentioning

confidence: 99%

“…An example is depicted in Fig. 2, and the labels express the stationary state satisfying the three conditions of Nahimovs, Rivosh, and Santos [10,11]. Since the initial uniform state ( 1) is approximately equal to this stationary state, the quantum algorithm is no better than classically guessing and checking.…”

Section: Introductionmentioning

confidence: 99%

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Stationary states in quantum walk search

2016

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When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if non-bipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

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Section: Introductionsupporting

confidence: 60%

Section: Introductioncontrasting

confidence: 51%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stationary states in quantum walk search

2016

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Exceptional quantum walk search on the cycle

Wong¹,

Santos²

2017

Quantum Inf Process

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Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random walk's hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then the speedup is roughly quadratic.

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Efficient quantum walk on the grid with multiple marked elements

Hoyer,

Komeili

2016

Preprint

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We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.

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Adjacent vertices can be hard to find by quantum walks

Cited by 3 publications

References 0 publications

Stationary states in quantum walk search

Stationary states in quantum walk search

Exceptional quantum walk search on the cycle

Efficient quantum walk on the grid with multiple marked elements

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