Time-Efficient Quantum Walks for 3-Distinctness [chapter]

Aleksandrs Belovs, Andrew M. Childs, Stacey Jeffery, Robin Kothari, Frédéric Magniez
2013 Lecture Notes in Computer Science  
We present two quantum walk algorithms for 3-Distinctness. Both algorithms have time complexityÕ(n 5/7 ), improving the previous O(n 3/4 ) and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors. The first algorithm is based on a connection between quantum walks and electric networks. The second algorithm uses an extension of the quantum walk search framework that facilitates quantum walks with nested updates. Introduction Element
more » ... s is a basic computational problem. Given a sequence χ = χ 1 , . . . , χ n of n integers, the task is to decide if those elements are pairwise distinct. This problem is closely related to Collision, a fundamental problem in cryptanalysis. Given a 2-to-1 function f : [n] → [n], the aim is to find a = b such that f (a) = f (b). One of the best (classical and quantum) algorithms is to run Element Distinctness on f restricted to a random subset of size √ n. In the quantum setting, Element Distinctness has received a lot of attention. The first non-trivial algorithm usedÕ(n 3/4 ) time [1] . The optimalÕ(n 2/3 ) algorithm is due to Ambainis [2], who introduced an approach based on quantum walks that has become a major tool for quantum query algorithms. The optimality of this algorithm follows from a query lower bound for Collision [3] . In the query model, access to the input χ is provided by an oracle whose answer to query i ∈ [n] is χ i . This model is the quantum analog of classical decision tree complexity: the only resource measured is the number of queries to the input. Quantum query complexity has been a very successful model for studying the power of quantum computation. In particular, bounded-error quantum query This paper is a merge of two submitted papers, whose full versions are available at
doi:10.1007/978-3-642-39206-1_10 fatcat:whevi2trfrbalgl4xayov3m2ei