The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm

TL;DR

This paper explores the fundamental quantum resource of magic in Shor's algorithm, linking it to the classical difficulty of factoring and providing a resource-based perspective on quantum computational costs.

Contribution

It develops an explicit analytic theory connecting quantum magic to the number-theoretic hardness in Shor's algorithm, highlighting the resource's role in quantum advantage.

Findings

Shor's algorithm maximally exploits quantum magic in relevant regimes.

A deep connection between classical problem difficulty and quantum resource cost is established.

The resource-based metric complements standard circuit cost analyses.

Abstract

The execution cost of quantum algorithms is typically quantified through asymptotic gate counts and qubit register sizes, yet these metrics do not directly capture which genuinely quantum resources, and in what amount, must be created and maintained for the computation to succeed. The systematic quantification of such information-theoretic requirements in quantum computing protocols remains an extremely challenging open problem, despite their direct role in establishing quantum advantage. To address this gap, we investigate the generation of non-stabilizerness (or magic), one of the key resources, in the paradigmatic Shor's factoring algorithm, revealing a deep connection between intrinsic quantum complexity and the computational hardness of the underlying number-theoretic problem. By developing an explicit analytic theory, we demonstrate the fundamental role of magic in the successful execution of the algorithm, and show that Shor's routine maximally exploits the quantum resource in practically relevant regimes. Our findings create a concise conceptual link between the classical algorithmic difficulty of a task and the non-stabilizer price to solve it on quantum hardware, complementing standard circuit-cost analyses with a resource-based metric that is naturally aligned with the real bottlenecks of fault-tolerant quantum computing.