The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N ≤ M), respectively, and the same range, the goal of the problem is to find x and y such that f (x) = g(y). This problem has been considered in both quantum and classical settings in t...

The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. ...

In this paper, we analyze the potential for new types of searches using the formalism of scattering random walks on Quantum Computers. Given a particular type of graph consisting of nodes and connections, a ”Tree Maze”, we would like to find a selected final node as quickly as possible, faster than any classical search algorithm. We show that this can be done using a quantum random walk, both e...

The Markovian random walk of a point at the velocity of fight on a two-dimensional invariant space-time lattice is shown to yield the quantum statistic associated with the Klein-Gordon equation. Quantum mechanics thus appem~ as a particular case of Markovian processes in velocity space: and one justifies the introduction of Dirac's invariant "ether" as a possible physical stochastic subquantum ...

Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As related results, an asymptotic description of Plancherel measure of the finite general linear groups is given, and a connection of these random walks with quantum computing is noted.

Extended abstract Many classical search problems can be cast in the following abstract framework: Given a finite set X and a subset M ⊆ X of marked elements, detect if M is empty or not, or find an element in M if there is any. When M is not empty, a naive approach to the finding problem is to repeatedly pick a uniformly random element of X until a marked element is sampled. A more sophisticate...

We show that the problem of Random Walk with boundary attractive potential may be mapped onto the free massive bosonic Quantum Field Theory with a line of defect. This mapping permits to recover the statistical properties of the Random Walks by using boundary S–matrix and Form Factor techniques. E-mail: [email protected]

Spatial search on graphs is one of the most important algorithmic applications quantum walks. To show that a quantum-walk-based more efficient than random-walk-based difficult problem, which has been addressed in several ways. Usually, graph symmetries aid calculation algorithm's computational complexity, and Johnson are an interesting class regarding because they regular, Hamilton-connected, v...