Optimized quantum random-walk search algorithms

TL;DR

This paper improves the quantum random-walk search algorithm by increasing success probability and optimizing query complexity, approaching the theoretical maximum efficiency for quantum search tasks.

Contribution

The authors enhance the SKW algorithm to achieve higher success probability and near-optimal query complexity, including adaptations for multiple marked elements.

Findings

Success probability significantly increased.

Query complexity optimized to near the theoretical limit.

Applicable to multiple marked elements.

Abstract

Shenvi, Kempe and Whaley's quantum random-walk search (SKW) algorithm [Phys. Rev. A 67, 052307 (2003)] is known to require $O(\sqrt N)$ number of oracle queries to find the marked element, where $N$ is the size of the search space. The overall time complexity of the SKW algorithm differs from the best achievable on a quantum computer only by a constant factor. We present improvements to the SKW algorithm which yield significant increase in success probability, and an improvement on query complexity such that the theoretical limit of a search algorithm succeeding with probability close to one is reached. We point out which improvement can be applied if there is more than one marked element to find.