Doubling the success of quantum walk search using internal-state measurements

Prūsis, Krišjānis; Vihrovs, Jevgēnijs; Wong, Thomas G.

doi:10.1088/1751-8113/49/45/455301

Cited by 9 publications

(14 citation statements)

References 26 publications

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“…Using (9) from [22], we can write the quantum walk operator U walk in this 4D subspace. Furthermore, since only |ab corresponds to a particle located at a marked vertex, the query is Q = diag{−1 1 1 1} in the 4D subspace.…”

Section: Marked Vertices In One Setmentioning

confidence: 99%

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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: Marked Vertices In One Setmentioning

confidence: 99%

“…In this basis, the quantum walk operator U walk = SC can be obtained using Eq. [9] from [22], and since both a and b vertices are marked, the oracle is Q = diag{−1, −1, −1, −1, 1, 1, 1, 1} in this basis. Combining them, the search operator U = U walk Q (1) is…”

Section: Marked Vertices In Both Setsmentioning

confidence: 99%

Quantum walk search on the complete bipartite graph

Rhodes

Wong

2019

Phys. Rev. A

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“…In particular, k = ω(N 2/3 ) means that Ω s,t = o(1) by (17) and therefore implies localization. For instance, edge-connectivity between any two vertices in the complete graph is high: Θ(N ) and then Ω s,t = O 1/N 1/3 .…”

Section: E High Connectivitymentioning

confidence: 99%

Oscillatory localization of quantum walks analyzed by classical electric circuits

et al. 2016

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We examine an unexplored quantum phenomenon we call oscillatory localization, where a discretetime quantum walk with Grover's diffusion coin jumps back and forth between two vertices. We then connect it to the power dissipation of a related electric network. Namely, we show that there are only two kinds of oscillating states, called uniform states and flip states, and that the projection of an arbitrary state onto a flip state is bounded by the power dissipation of an electric circuit. By applying this framework to states along a single edge of a graph, we show that low effective resistance implies oscillatory localization of the quantum walk. This reveals that oscillatory localization occurs on a large variety of regular graphs, including edge-transitive, expander, and high degree graphs. As a corollary, high edge-connectivity also implies localization of these states, since it is closely related to electric resistance.

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“…Then the oracle/coin −C in U and −I in U act identically on the marked vertex. This is explicitly proved in Section 2 of [13], and after π √ N /2 √ 2 applications of either operator, the system achieves a success probability of 1/2 when measuring the position of the particle (although an internal-state measurement can further improve this [14]). As another example, the two operators U and U are equivalent for search on arbitrary-dimensional periodic square lattices with a single marked vertex [6].…”

Section: Skw Oraclementioning

confidence: 87%

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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Doubling the success of quantum walk search using internal-state measurements

Cited by 9 publications

References 26 publications

Quantum walk search on the complete bipartite graph

Quantum walk search on the complete bipartite graph

Oscillatory localization of quantum walks analyzed by classical electric circuits

Stationary states in quantum walk search

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