Adjacent vertices can be hard to find by quantum walks

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω( N ) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω( N ) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph. Additionally, we consider an algorithmic application of the found effect. We investigate a problem of detection of a perfect matching in a bipartite graph. We use the found effect as an algorithmic building block and construct quantum algorithm which, for a specific class of graphs, outperforms its classical analogs.

arXiv:1605.05598Moscow University Computational Mathematics and CyberneticsJan 1, 20196 citations

Quantum-walk-based search algorithms with multiple marked vertices

Gustavo Alves Bezerra, Pedro H. G. Lugão, Renato Portugal

The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.

arXiv:2103.12878Physical Review AJun 1, 202122 citations

Quantum Walks on Two-Dimensional Grids with Multiple Marked Locations

Nikolajs Nahimovs, Alexander Rivosh

The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration (the placement and the number) of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [3]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for several configurations of multiple 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 [Formula: see text] marked locations for which the number of steps of the algorithm differs by a factor of [Formula: see text]. Second, we present two configurations of [Formula: see text] and [Formula: see text] marked locations having the same number of steps and probability of finding a marked location.The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration (the placement and the number) of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [3]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for several configurations of multiple 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 [Formula: see text] marked locations for which the number of steps of the algorithm differs by a factor of [Formula: see text]. Second, we present two configurations of [Formula: see text] and [Formula: see text] marked locations having the same number of steps and probability of finding a marked location.

International Journal of Foundations of Computer ScienceJun 1, 20183 citations

Lackadaisical quantum walks with multiple marked vertices

Nikolajs Nahimovs

The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.

arXiv:1808.00672Aug 2, 20180 citations

Quantum walks on two-dimensional grids with multiple marked locations

Nikolajs Nahimovs, Alexander Rivosh

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 $k$ marked locations for which the number of steps of the algorithm differs by $Ω(\sqrt{k})$ factor. Second, we present two configurations of $k$ and $\sqrt{k}$ marked locations having the same number of steps and probability to find a marked location.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 $k$ marked locations for which the number of steps of the algorithm differs by $Ω(\sqrt{k})$ factor. Second, we present two configurations of $k$ and $\sqrt{k}$ marked locations having the same number of steps and probability to find a marked location.

arXiv:1507.03788Jul 14, 20153 citations

Optimality conditions for spatial search with multiple marked vertices

Mathieu Roget, Hachem Kadri, Giuseppe Di Molfetta

We contribute to fulfill the long-lasting gap in the understanding of the spatial search with multiple marked vertices. The theoretical framework is that of discrete-time quantum walks (QW), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. QW based search algorithms are well understood when they have to tackle the fundamental problem of finding only one marked element in a $d\text{\ensuremath{-}}\text{dimensional}$ grid and it has been proven they provide a quadratic advantage over classical searching protocols. However, once we consider searching more than one element, the behavior of the algorithm may be affected by the spatial configuration of the marked elements and even the quantum advantage is no longer guaranteed. Here our main contribution is threefold: (i) we provide sufficient conditions for optimality for a multi-items QWSearch algorithm; (ii) we provide analytical evidence that almost, but not all spatial configurations with multiple marked elements are optimal; and (iii) we numerically show that the computational advantage with respect to the classical counterpart is not always certain and it does depend on the proportion of searched elements over the total number of grid points.We contribute to fulfill the long-lasting gap in the understanding of the spatial search with multiple marked vertices. The theoretical framework is that of discrete-time quantum walks (QW), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. QW based search algorithms are well understood when they have to tackle the fundamental problem of finding only one marked element in a $d\text{\ensuremath{-}}\text{dimensional}$ grid and it has been proven they provide a quadratic advantage over classical searching protocols. However, once we consider searching more than one element, the behavior of the algorithm may be affected by the spatial configuration of the marked elements and even the quantum advantage is no longer guaranteed. Here our main contribution is threefold: (i) we provide sufficient conditions for optimality for a multi-items QWSearch algorithm; (ii) we provide analytical evidence that almost, but not all spatial configurations with multiple marked elements are optimal; and (iii) we numerically show that the computational advantage with respect to the classical counterpart is not always certain and it does depend on the proportion of searched elements over the total number of grid points.

arXiv:2201.12937Physical Review ResearchJul 11, 20234 citations

k

marked locations for which the number of steps of the algorithm differs by

Ω(\sqrt{k})

factor. Second, we present two configurations of

k

and

\sqrt{k}

marked locations having the same number of steps and probability to find a marked location.