Time-Efficient Quantum Walks for 3-Distinctness

Belovs, Aleksandrs; Childs, Andrew M.; Jeffery, Stacey; Kothari, Robin; Magniez, Frédéric

doi:10.1007/978-3-642-39206-1_10

Cited by 33 publications

(36 citation statements)

References 21 publications

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“…For quantum walks, however, such connections are relatively new. Belovs et al [16] bounds the running time of a quantum walk algorithm for 3-Distinctness in terms of the resistance of a graph. We begin with a simple example of oscillatory localization on the complete graph of N vertices, an example of which is shown in Fig.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

Oscillatory localization of quantum walks analyzed by classical electric circuits

et al. 2016

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“…Using Eq. (18), we see that the only free parameter in b is t 2 . So, we choose t 2 that minimizes b. Eqs.…”

Section: Resultsmentioning

confidence: 75%

Element distinctness revisited

Portugal

2018

Quantum Inf Process

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The element distinctness problem is the problem of determining whether the elements of a list are distinct, that is, if x = (x 1 , ..., x N ) is a list with N elements, we ask whether the elements of x are distinct or not. The solution in a classical computer requires N queries because it uses sorting to check whether there are equal elements. In the quantum case, it is possible to solve the problem in O(N 2/3 ) queries. There is an extension which asks whether there are k colliding elements, known as element k-distinctness problem.This work obtains optimal values of two critical parameters of Ambainis' seminal quantum algorithm [SIAM J. Comput., 37, 210-239, 2007]. The first critical parameter is the number of repetitions of the algorithm's main block, which inverts the phase of the marked elements and calls a subroutine. The second parameter is the number of quantum walk steps interlaced by oracle queries. We show that, when the optimal values of the parameters are used, the algorithm's success probability is 1 − O(N 1/(k+1) ), quickly approaching 1. The specification of the exact running time and success probability is important in practical applications of this algorithm. * portugal@lncc.br

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“…Thus, the probability that Algorithm 1 does not exit before querying an η corresponding to a β η such that |β η − π/4| ≤ π/8 is 1 − O(δ 0 /n). 3 Similarly, the probability that the algorithm exits once it reaches a state for which |β η − π/4| ≤ π/8 is 1 − O((δ 0 /n). The probability of success is then Now, for the full sequence of queries (…”

Section: Failure Ratementioning

confidence: 98%

Improved quantum backtracking algorithms using effective resistance estimates

Jarret

Wan

2018

Phys. Rev. A

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We investigate quantum backtracking algorithms of the type introduced by Montanaro (arXiv:1509.02374). These algorithms explore trees of unknown structure and in certain settings exponentially outperform their classical counterparts. Some of the previous work focused on obtaining a quantum advantage for trees in which a unique marked vertex is promised to exist. We remove this restriction by recharacterising the problem in terms of the effective resistance of the search space. In this paper, we present a generalisation of one of Montanaro's algorithms to trees containing k marked vertices, where k is not necessarily known a priori.Our approach involves using amplitude estimation to determine a near-optimal weighting of a diffusion operator, which can then be applied to prepare a superposition state with support only on marked vertices and ancestors thereof. By repeatedly sampling this state and updating the input vertex, a marked vertex is reached in a logarithmic number of steps. The algorithm thereby achieves the conjectured bound of O( √ T R max ) for finding a single marked vertex and O k √ T R max for finding all k marked vertices, where T is an upper bound on the tree size and R max is the maximum effective resistance encountered by the algorithm. This constitutes a speedup over Montanaro's original procedure in both the case of finding one and the case of finding multiple marked vertices in an arbitrary tree.

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Time-Efficient Quantum Walks for 3-Distinctness

Cited by 33 publications

References 21 publications

Oscillatory localization of quantum walks analyzed by classical electric circuits

Oscillatory localization of quantum walks analyzed by classical electric circuits

Element distinctness revisited

Improved quantum backtracking algorithms using effective resistance estimates

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