Span-program-based quantum algorithm for the rank problem

Recently, span programs have been shown to be equivalent to quantum query algorithms. It is an open problem whether this equivalence can be utilized in order to come up with new quantum algorithms. We address this problem by providing span programs for some linear algebra problems. We develop a notion of a high level span program, that abstracts from loading input vectors into a span program. Then we give a high level span program for the rank problem. The last section of the paper deals with reducing a high level span program to an ordinary span program that can be solved using known quantum query algorithms.