1805.02714May 7, 201841 pages, 2 figures. Changes in definition of non-binary span program and learning graph leading to major improvements in the results. Arguments added to support the naturalness of the definition of non-binary span program in the text and in a new appendix
Span programs characterize the quantum query complexity of binary functions f:{0,…,ℓ}n→{0,1} up to a constant factor. In this paper we generalize the notion of span programs for functions with non-binary input/output alphabets f:[ℓ]n→[m]
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. We show that non-binary span program characterizes the quantum query complexity of any such function up to a constant factor. We argue that this non-binary span program is indeed the generalization of its binary counterpart. We also generalize the notion of span programs for a special class of relations. Learning graphs provide another tool for designing quantum query algorithms for binary functions. In this paper, we also generalize this tool for non-binary functions, and as an application of our non-binary span program show that any non-binary learning graph gives an upper bound on the quantum query complexity.