Quantum Algorithms for Lowest Weight Paths and Spanning Trees in Complete Graphs

Quantum algorithms for several problems in graph theory are considered. Classical algorithms for finding the lowest weight path between two points in a graph and for finding a minimal weight spanning tree involve searching over some space. Modification of classical algorithms due to Dijkstra and Prim allows quantum search to replace classical search and leads to more efficient algorithms. In the case of highly asymmetric complete bipartite graphs, simply replacing classical search with quantum search leads to a faster quantum algorithm. A fast quantum algorithm for computing the diameter of a complete graph is also given. Introduction The question of which classical algorithms can be sped up by quantum computing is of course a very interesting one. At present there are only a few general techniques known in the field of quantum computing and finding new problems that are amenable to quantum speedups is a high priority. Classically, one area of mathematics that is full of interesting algorithms is computational graph theory. It is therefore natural to ask whether any of the classical graph theory algorithms can take advantage of quantum computing. One of the few general techniques known centers around Grover’s algorithm for searching an unsorted list for a specified element. This original idea has been extended to general amplitude amplification that can be applied to any classical algorithm. It would be incorrect to assume that amplitude amplification always leads to quantum speedups of classical algorithms. There are some interesting cases where “Grover-like” techniques do in fact lead to speedups of classical algorithms. One very important case of this is to find the minimum value of a computable function as the set of input arguments ranges over a finite, but unordered list. In this case, if the list is of length n, then the quantum cost of finding the minimum is O (√ n ) , while the classical cost is O (