# An exploration of the noise sensitivity of the Shor's algorithm

## TL;DR

This paper investigates the noise sensitivity of Shor's quantum algorithm, revealing its superior fault tolerance under Z noise and providing a predictive model for large-scale factoring under biased noise conditions.

## Contribution

It demonstrates inherent noise resilience in Shor's algorithm, especially under Z noise, and introduces an extrapolation method to estimate success probabilities for large integers.

## Key findings

- Shor's algorithm shows better fault tolerance under Z noise than X and Y noise.
- Fault-tolerant positions grow quartically with problem size under Z noise.
- Estimated success probability for factoring 2048-bit integers under biased noise is approximately 1.417*10^{-17}.

## 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}.

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Source: https://tomesphere.com/paper/2509.00417