Span Program for Non-binary Functions
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Span programs characterize the quantum query complexity of binary functions f:{0,1}^n →{0,1} up to a constant factor. In this paper we generalize the notion of span programs for functions with non-binary input and/or output alphabets f:[ℓ]^n → [m]. We show that for any non-binary span program for such a function f with complexity C, the quantum query complexity of f is at most Q(f)=O(C). Conversely, there exists a non-binary span program for f with complexity √(ℓ-1) Q(f). Thus, we conclude that non-binary span programs characterize the quantum query complexity of f up to a factor of order at most √(ℓ-1). By giving explicit examples, we show that this √(ℓ-1) factor cannot be improved. We also generalize the notion of span programs for a special class of relations and prove similar results. 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.
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