Quantum Computation in Computational Geometry

Next, as a showcase problem of our approach, we consider the problem of finding the lowest point in the intersection of n upper-halfspaces in the d-dimensional Euclidean space. The problem can be solved in �(n) time if d is a constant in RAM model. Regarding the problem as a parametric minimax problem [2], and by combining the quantum minimum finding algorithm and multidimensional searching technique, we can solve it in O( p nlog 2d 1 n) time. Then, we show some other geometric optimization problems such as the minimum enclosing ball problem and linear programming problems can be solved efficiently in quantum model if d is a constant. We can regard the minimum enclosing ball problem as a minimization of the parametric farthest-neighbor-query problem [2]; in other words, it is computation of the lowest point in the upper envelope of surfaces defined from the distance functions from points. Then it can be solved in O( p nlog 2d+1 n) time. Also, some output-sensitive convex-hull algorithms and the ellipsoid algorithm for solving a general dimensional convex programming problem can be accelerated by using the quantum model.