Optimal Separation in Exact Query Complexities for Simon's Problem

TL;DR

This paper presents a simple, optimal quantum algorithm for Simon's problem with O(n) queries, establishing the best possible separation between quantum and classical exact query complexities.

Contribution

The paper introduces a new, straightforward exact quantum algorithm for Simon's problem that achieves the optimal separation in query complexities.

Findings

Quantum algorithm solves Simon's problem with O(n) queries.

Classical deterministic algorithm requires O(√2^n) queries.

Established the optimal separation as Θ(n) versus Θ(√2^n).

Abstract

Simon's problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simon's discussion on his problem was limited to bounded-error setting, which means his algorithm can not always get the correct answer. Exact quantum algorithms for Simon's problem have also been proposed, which deterministically solve the problem with O(n) queries. Also the quantum lower bound \Omega(n) for Simon's problem is known. Although these algorithms are either complicated or specialized, their results give an O(n) versus \Omega(\sqrt{2^{n}}) separation in exact query complexities for Simon's problem (\Omega(\sqrt{2^{n}}) is the lower bound for classical probabilistic algorithms), but it has not been proved whether this separation is optimal. In this paper, we propose another exact quantum algorithm for solving Simon's problem with O(n) queries, which is simple, concrete and does not rely on special query oracles. Our algorithm combines Simon's algorithm with the quantum amplitude amplification technique to ensure its determinism. In particular, we show that Simon's problem can be solved by a classical deterministic algorithm with O(\sqrt{2^{n}}) queries (as we are aware, there were no classical deterministic algorithms for solving Simon's problem with O(\sqrt{2^{n}}) queries). Combining some previous results, we obtain the optimal separation in exact query complexities for Simon's problem: \Theta({n}) versus \Theta({\sqrt{2^{n}}}).