Quantum search with advice

TL;DR

This paper introduces a quantum algorithm for searching an unstructured list with advice, achieving optimal query complexity and exponential speed-ups for certain distributions, improving over classical methods.

Contribution

It presents a quantum search algorithm that optimally uses advice distributions, including a variant for unknown distributions, based on Grover's search and amplitude amplification.

Findings

Achieves optimal query complexity up to a constant factor.

Exponential speed-ups for specific distributions like power laws.

Efficient algorithm for unknown advice distributions with additional querying.

Abstract

We consider the problem of search of an unstructured list for a marked element, when one is given advice as to where this element might be located, in the form of a probability distribution. The goal is to minimise the expected number of queries to the list made to find the marked element, with respect to this distribution. We present a quantum algorithm which solves this problem using an optimal number of queries, up to a constant factor. For some distributions on the input, such as certain power law distributions, the algorithm can achieve exponential speed-ups over the best possible classical algorithm. We also give an efficient quantum algorithm for a variant of this task where the distribution is not known in advance, but must be queried at an additional cost. The algorithms are based on the use of Grover's quantum search algorithm and amplitude amplification as subroutines.