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arXiv:2201.12937 · Physical Review Research

Optimality conditions for spatial search with multiple marked vertices

Mathieu Roget, Hachem Kadri, Giuseppe Di Molfetta

Published Jul 11, 2023• 4 citations
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Abstract

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.

DOI
10.1103/physrevresearch.5.033021

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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\ensuremath-dimensionald\text{\ensuremath{-}}\text{dimensional}d\ensuremath-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.
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.
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.
One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of nnn nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in O(n)O(\sqrt{n})O(n​) time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of the necessary and sufficient conditions that a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than ${n}^{\ensuremath{-}1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide examples of graphs where quantum search is suboptimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.
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.
Search algorithms based on discrete-time quantum walk (QW) can be considered as alterations of the standard QW: Use a different coin operator that distinguishes target and nontarget vertices, or, mark the target vertices first followed by the standard QW. Two most frequently used marking coins are [Formula: see text] and [Formula: see text] the negative identity operator and the negative Grover diffusion operator. We show that search algorithms corresponding to these four combinations can be reduced to two, denoted as [Formula: see text] and [Formula: see text], and they are equivalent when searching for nonadjacent multiple targets. For adjacent target vertices, numerical simulations show that the performance of the algorithm [Formula: see text] highly depends on the density of the underlying graph, and it outperforms [Formula: see text] when the density is large enough. At last, a generalized stationary state of both search algorithms on the graphs with even-numbered degree is provided.
The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is O(N)O(\sqrt {N})O(N​) with an asymptotic probability of 1+ o (1), where N is the number of vertices.