Improving the query complexity of quantum spatial search in two dimensions

TL;DR

This paper improves the efficiency of quantum spatial search in two dimensions by optimizing oracle complexity through a novel multi-step quantum walk approach that avoids amplitude amplification.

Contribution

It introduces a new method to make the oracle complexity optimal in 2D quantum spatial search using graph powering and multi-step quantum walks.

Findings

Oracle complexity is made optimal with a logarithmic increase in walk operator calls.

The algorithm implements multi-step quantum walks via graph powering.

Methods are applicable to quantum walks from powers of symmetric Markov chains.

Abstract

The question of whether quantum spatial search in two dimensions can be made optimal has long been an open problem. We report progress towards its resolution by showing that the oracle complexity for target location can be made optimal, by increasing the number of calls to the walk operator that incorporates the graph structure by a logarithmic factor. Our algorithm does not require amplitude amplification. An important ingredient of our algorithm is the implementation of multi-step quantum walks by graph powering, using a coin space of walk-length dependent dimension, which may be of independent interest. Finally, we demonstrate how to implement quantum walks arising from powers of symmetric Markov chains using our methods.