On the Quantum Black-Box Complexity of Majority [article]

Thomas Hayes, Samuel Kutin, Dieter van Melkebeek
2002 arXiv   pre-print
We describe a quantum black-box network computing the majority of N bits with zero-sided error eps using only 2N/3 + O(sqrtN (log log N + log 1/eps)) queries: the algorithm returns the correct answer with probability at least 1 - eps, and "I don't know" otherwise. Our algorithm is given as a randomized "XOR decision tree" for which the number of queries on any input is strongly concentrated around a value of at most 2N/3. We provide a nearly matching lower bound of 2N/3 - O(sqrt(N)) on the
more » ... ted number of queries on a worst-case input in the randomized XOR decision tree model with zero-sided error o(1). Any classical randomized decision tree computing the majority on N bits with zero-sided error 1/2 has cost N.
arXiv:quant-ph/0109101v3 fatcat:iyy6fwqkvnhf5iubi4yudjs65u