Spatial search by quantum walk

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of N items laid out in d spatial dimensions can be searched in time of order sqrt(N) for d>2, and in time of order sqrt(N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous time quantum walk on a graph. The case of the complete graph gives the continuous time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that sqrt(N) speedup can also be achieved on the hypercube. We show that full sqrt(N) speedup can be achieved on a d-dimensional periodic lattice for d>4. In d=4, the quantum walk search algorithm takes time of order sqrt(N) poly(log N), and in d<4, the algorithm does not provide substantial speedup.Comment: v2: 12 pages, 4 figures; published version, with improved arguments for the cases where the algorithm fail

Hybrid Variable Neighborhood HyperHeuristics for Exam Timetabling Problems

This paper presents our work on analysing the high level search within a graph based hyperheuristic. The graph based hyperheuristic solves the problem at a higher level by searching through permutations of graph heuristics rather than the actual solutions. The heuristic permutations are then used to construct the solutions. Variable Neighborhood Search, Steepest Descent, Iterated Local Search and Tabu Search are compared. An analysis of their performance within the high level search space of heuristics is also carried out. Experimental results on benchmark exam timetabling problems demonstrate the simplicity and efficiency of this hyperheuristic approach. They also indicate that the choice of the high level search methodology is not crucial and the high level search should explore the heuristic search space as widely as possible within a limited searching time. This simple and general graph based hyperheuristic may be applied to a range of timetabling and optimisation problems