There exist two types of configurations of marked vertices on a two-dimensional grid, known as the exceptional configurations, which are hard to find by the discrete-time quantum walk algorithms. In this article, we provide a comparative study of the quantum walk algorithm with different coins to search these exceptional configurations on a two-dimensional grid. We further extend the analysis to the hypercube, where only one type of exceptional configurations are present. Our observation, backed by numerical results, is that our recently proposed modified coin operator is the only coin which can search both types of exceptional configurations as well as nonexceptional configurations successfully. As a consequence, we observe that the existence of exceptional configurations are not a quantum phenomenon, rather a mere limitation of some of the coin operators.
There exist two types of configurations of marked vertices on a two-dimensional grid, known as the exceptional configurations, which are hard to find by the discrete-time quantum walk algorithms. In this article, we provide a comparative study of the quantum walk algorithm with different coins to search these exceptional configurations on a two-dimensional grid. We further extend the analysis to the hypercube, where only one type of exceptional configurations are present. Our observation, backed by numerical results, is that our recently proposed modified coin operator is the only coin which can search both types of exceptional configurations as well as nonexceptional configurations successfully. As a consequence, we observe that the existence of exceptional configurations are not a quantum phenomenon, rather a mere limitation of some of the coin operators.
Abstract We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Grover coin. We find that the most common coin combination model (G, −), where G is a Grover diffusion transformation, is a generalized exceptional configuration when just searching one marked vertex on the cycle. In the end, we find generalized exceptional configuration has a different evolution of quantum coherence from exceptional configuration. These extend largely the range of exceptional configuration of quantum walk search in some sense.
There exist several types of configurations of marked vertices, referred to as the exceptional configurations, on one- and two-dimensional periodic lattices with additional long-range edges of the Hanoi network of degree four (HN4), which are challenging to find using discrete-time quantum walk algorithms. In this article, we conduct a comparative analysis of the discrete-time quantum walk algorithm utilizing various coin operators to search for these exceptional configurations. First, we study the emergence of several new exceptional configurations/vertices due to the additional long-range edges of the HN4 on both one- and two-dimensional lattices. Second, our study shows that the diagonal configuration on a two-dimensional lattice, which is exceptional in the case without long-range edges, no longer remains an exceptional configuration. Third, it is also shown that a recently proposed modified coin can search all these configurations, including any other configurations in one- and two-dimensional lattices with very high success probability. Additionally, we construct stationary states for the exceptional configurations caused by the additional long-range edges, which explains why the standard and lackadaisical quantum walks with the Grover coin cannot search these configurations.
Quantum walk search on graphs is primarily limited to real evolution operators, although there is no restriction for using the complex unitary matrix as the evolution operator. In the context of quantum search, several studies on the Grover’s search utilize the complex unitary matrix as an evolution operator, which allows for efficient searching of marked elements. However, the generalization of the quantum walk search to complex space has largely been overlooked. In this paper, we extend existing discrete-time coined quantum walk search algorithms on one- and two-dimensional periodic lattices to complex space by introducing two phase parameters, [Formula: see text], and [Formula: see text], into the coin and oracle operators, respectively, for four different discrete-time coined quantum walk search algorithms. One of the advantages of this generalization is that it enables the Grover and SKW coins with specific values of the phase parameters [Formula: see text], and [Formula: see text] to search for the exceptional configurations on a two-dimensional periodic lattice with high success probability, which is otherwise impossible to search using the real Grover and SKW coins operators.
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 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.
In seminal work, Szegedy gave a method for quantizing a classical Markov chain that has been especially useful for searching for marked vertices in graphs. We prove, however, that this algorithm fails when searching on the two-dimensional periodic square lattice with a marked diagonal, in the sense that the state of the system only evolves by acquiring minus signs. Then the state is a uniform probability distribution over the vertices for all time, which is equivalent to classically guessing and checking. Furthermore, we prove that this failure also occurs for any configuration of marked vertices on the one-dimensional cycle and for any configuration that reduces to it. Thus, Szegedy's quantum walk has a family of exceptional configurations for which it does not spread. Despite this, we exploit this stationary behavior to construct a quantum walk with an arbitrary polynomial speedup over the classical random walk, and it is the first example of a greater-than-quadratic speedup.