📝 SummarXiv

Abstract

Quantum algorithms face significant challenges due to qubit susceptibility to environmental noise, and quantum error correction typically requires prohibitive resource overhead. This paper proposes that quantum algorithms may possess inherent noise resilience characteristics that could reduce implementation barriers. We investigate Shor's algorithm by applying circuit-level noise models directly to the original algorithm circuit. Our findings reveal that Shor's algorithm demonstrates superior fault tolerance under Z noise compared to X and Y noise. Focusing on the modular exponentiation circuit which is the core component of the algorithm, we conduct fault-tolerant position statistics on circuits with bit lengths from 4 to 9. The results show that under Z noise, fault-tolerant positions grow with the same quartic polynomial order as potential error positions as the problem scale increases. In contrast, fault tolerance under X and Y noise exhibits a strong dependence on the composite number N and the parameter a. Based on these findings, we develop an extrapolation method predicting that the minimum probability of a correct output of the modular exponentiation circuit to factor 2048 bit integers under biased noise is approximately 1.417*{10}^{-17}.