Finding paths with quantum walks or quantum walking through a maze

Abstract

We show that it is possible to use a quantum walk to find a path from one marked vertex to another. In the specific case of M stars connected in a chain, one can find the path from the first star to the last one in O(M √ N ) steps, where N is the number of spokes of each star. First we provide an analytical result showing that by starting in a phase-modulated highly superposed initial state we can find the path in O(M √ N log M ) steps. Next, we improve this efficiency by showing that the recovery of the path can also be performed by a series of successive searches when we start at the last known position and search for the next connection in O( √ N ) steps leading to the overall efficiency of O(M √ N ). For this result we use the analytical solution that can be obtained for a ring of stars of double the length of the chain.

Maze Solving by a Quantum Walk with Sinks and Self-Loops: Numerical Analysis

“…The key features of the proposed method are the self-loops at the start and the goal and the sink node attached to the goal. In previous studies, the start and goal were marked by reflection with phase inversion placed at the dead ends [18][19][20]. The correct path was then judged by the transient profile of the probabilities.…”

“…In this paper, we numerically examine a maze-solving method that uses a quantum walk on a network. The presented method is an application of the emergence of a trapped eigenstate on a network with sinks, and it provides an alternative to previously reported methods [18][19][20]. Although the mathematical foundation of this method was given by Konno, Segawa, and Štefa ňák [28], the results presented here are non-trivial because the interaction among multiple trapped eigenstates and the initial condition is generally difficult to characterize as of now.…”

“…Maze-solving using quantum walks has been studied by Hillery, Koch, and Reitzner on an N-tree maze [18] and a chain of stars [19,20]. Their works are the extension of their studies on quantum search and finding structural anomalies in networks [21][22][23][24][25][26][27].…”

Scattering quantum random walks on square grids and randomly generated mazes

“…Lastly, we show the extent to which the SQRW scheme is still viable in scenarios with randomness in both the location of the special vertex and the geometry of the system itself. In all search algorithms the location of F is always unknown, but here we study a more difficult case where this randomness directly affects the optimal way to prepare the quantum system, unlike [25,26]. We also show how the SQRW scheme fairs under conditions where the searching geometry has unknown barriers placed throughout.…”

“…In more recent studies, it has been show that it is possible to use Scattering Quantum Random Walks (SQRW), a particular type of quantum random walk scheme, to create probability distributions that can aid in finding a marked vertex [25,26]. In both cases, quantum speedups were found as a result of the SQRW causing nearly all of the probability in the systems to concentrate along the path of states leading to F. This paper is an extension to the study of SQRWs, showcasing a new geometry that one can obtain a speedup on.…”