On the Role of Coherence in Shor's Algorithm

TL;DR

This paper investigates how quantum coherence influences the success of a structured variant of Shor's algorithm, establishing bounds on its performance and highlighting coherence as a key quantum resource.

Contribution

It introduces a framework using dynamical resource theories to quantify coherence's role in a fixed-structure Shor's algorithm, providing bounds on success probability.

Findings

Coherence bounds success probability of the algorithm.

Quantum coherence is identified as the key resource for the algorithm's performance.

The study links coherence measures to the algorithm's effectiveness.

Abstract

Shor's factoring algorithm provides a super-polynomial speed-up over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depend on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.