Practical Challenges in Executing Shor's Algorithm on Existing Quantum Platforms

TL;DR

This paper evaluates practical barriers to executing Shor's algorithm on current quantum platforms by surveying qubit technologies, reviewing implementation progress, and performing order-finding experiments on an IBM superconducting device. It demonstrates a persistent gap between hardware capabilities and cryptographically relevant factoring due to architecture-specific circuit design and unstable fidelities, despite significant algorithmic and hardware advances. The authors discuss the challenges of generic, platform-agnostic Shor implementations, report small-modulus experimental results, and project that cryptanalytic-scale factoring will require substantial, multi-year progress in qubit counts and error correction. They argue for ongoing, evidence-based monitoring of quantum capabilities to inform cryptographic preparedness and policy planning.

Abstract

Quantum computers pose a fundamental threat to widely deployed public-key cryptosystems, such as RSA and ECC, by enabling efficient integer factorization using Shor's algorithm. Theoretical resource estimates suggest that 2048-bit RSA keys could be broken using Shor's algorithm with fewer than a million noisy qubits. Although such machines do not yet exist, the availability of smaller, cloud-accessible quantum processors and open-source implementations of Shor's algorithm raises the question of what key sizes can realistically be factored with today's platforms. In this work, we experimentally investigate Shor's algorithm on several cloud-based quantum computers using publicly available implementations. Our results reveal a substantial gap between the capabilities of current quantum hardware and the requirements for factoring cryptographically relevant integers. In particular, we observe that circuit constructions still need to be highly specific for each modulus, and that machine fidelities are unstable, with high and fluctuating error rates.

Figures (5)

• Figure 1: Measured QPE histogram for $N=15$ (simulation).
• Figure 2: Measured QPE histogram for $N=15$.
• Figure 3: Measured QPE histogram for $N=21$.
• Figure 4: Measured QPE histogram for $N=35$$(a=4)$.
• Figure 5: Measured QPE histogram for $N=35$$(a=8)$.