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Algorithms

Searching via Nonlinear Quantum Walk on the 2D-Grid

Giuseppe Di Molfetta, Basile Herzog

Published Nov 23, 2020• 4 citations
Publisher

Abstract

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.

DOI
10.3390/a13110305

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We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log N)$, where $N$ is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum walks.

arXiv:1312.0172•Dec 1, 2013•2 citations
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage \BHg{with} respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \cite{Childs_2014}. The numerical simulations showed that the walker finds the marked vertex in $O(N^{1/4} \log^{3/4} N) $ steps, with probability O(1/log⁡N)O(1/\log N)O(1/logN), for an overall complexity of O(N1/4log⁡7/4N)O(N^{1/4}\log^{7/4}N)O(N1/4log7/4N). We also proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm.
We provide first evidence that some families of nonlinear quantum systems, rephrased in terms of coined quantum walks with effective nonlinear phase, display a strong computational advantage for search algorithms, over graphs having sets of vertices of constant size, e.g. the infinite square grid. The numerical simulations show that the walker finds the marked vertex in O(N1/4log⁡3/4N)O(N^{1/4} \log^{3/4} N)O(N1/4log3/4N) steps with probability O(1/log⁡N)O(1/\log N)O(1/logN), for an overall complexity of O(N1/4log⁡7/4N)O(N^{1/4}\log^{7/4}N)O(N1/4log7/4N). We also confirm this result analytically. Moreover the algorithm is implemented without the need for a specific oracle step, but by topological hole defect in the grid. From a quantum computing perspective, however, this hints at novel applications of quantum walk search: instead of using them to look for good solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space.
The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration of marked locations. While the first dependence have been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two marked location cases only. We move beyond two marked locations and study the behaviour of the algorithm for an arbitrary configuration of marked locations. In this paper we prove two results showing the importance of how the marked locations are arranged. First, we present two placements of kkk marked locations for which the number of steps of the algorithm differs by Ω(k)Ω(\sqrt{k})Ω(k​) factor. Second, we present two configurations of kkk and k\sqrt{k}k​ marked locations having the same number of steps and probability to find a marked location.
n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter ϵ\epsilonϵ. More specifically, for an NNN-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps hhh satisfies h≥ln⁡(2ϵ)Nh \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}h≥ln(ϵ​2​)N​, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]\epsilon\in (0,1]ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size Θ(N)Θ(N)Θ(N), for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.
We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is O(Nlog⁡N)O(\sqrt{N\log N})O(NlogN​) and the success probability is O(1/log⁡N)O(1/\log N)O(1/logN), where NNN is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum walks.