The Quantum Black-Box Complexity of Majority

We describe a quantum black-box network computing the majority of N bits with zerosided error using only 2 3 N + O( N log( -1 log N)) queries: the algorithm returns the correct answer with probability at least 1 - , 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 2 3 N. We provide a nearly matching lower bound of 2 3 N - O( N) on the expected 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.

• Type: Journal
• Year: 2002
• Where: ALGORITHMICA
• Authors: Thomas P. Hayes, Samuel Kutin, Dieter van Melkebeek
• Keywords: ALGORITHMICA 2002, Quantum Black-box Network, Randomized Xor Decision, XOR Decision Tree