Information-theoretic lower bounds for quantum sorting

TL;DR

This paper investigates the quantum query complexity of sorting with partial information, demonstrating that for many partial orders, quantum algorithms do not significantly outperform classical bounds.

Contribution

It extends previous results by proving quantum lower bounds for a broad class of partially ordered sets, improving upon earlier work by Yao.

Findings

Quantum query complexity matches classical lower bounds for many partial orders.

The result applies to a wide class of partially ordered sets.

Quantum advantage in sorting with partial information is limited.

Abstract

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).