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Bounds on quantum ordered searching

Peter Hoyer·Jan Neerbek
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Abstract

We prove that any exact quantum algorithm searching an ordered list of N elements requires more than \frac{1}π(\ln(N)-1) queries to the list. This improves upon the previously best known lower bound of {1/12}\log_2(N) - O(1). Our proof is based on a weighted all-pairs inner product argument, and it generalizes to bounded-error quantum algorithms. The currently best known upper bound for exact searching is roughly 0.526 \log_2(N). We give an exact quantum algorithm that uses \log_3(N) + O(1) queries, which is roughly 0.631 \log_2(N). The main principles in our algorithm are an quantum parallel use of the classical binary search algorithm and a method that allows basis states in superpositions to communicate.

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Peter Hoyer, Jan Neerbek. 2001-04-27. Bounds on quantum ordered searching. https://arxiv.org/abs/quant-ph/0009032

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