Quantum Complexity of Testing Group Commutativity

TL;DR

This paper presents a quantum algorithm for testing group commutativity with complexity O(k^{2/3}), improving upon previous randomized algorithms, and establishes matching lower bounds using reductions from Element Distinctness.

Contribution

The paper introduces a nearly optimal quantum algorithm for testing group commutativity and proves its optimality through lower bounds, demonstrating the power of quantum techniques in group theory problems.

Findings

Quantum algorithm achieves O(k^{2/3}) complexity.

Lower bound matches the quantum algorithm complexity.

Optimality of previous randomized algorithms is established.

Abstract

We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.