Shor's Algorithm Does Not Factor Large Integers in the Presence of Noise

TL;DR

This paper demonstrates that Shor's quantum factoring algorithm fails to factor large integers when subjected to realistic noise levels in quantum gates, challenging its practical viability under current quantum hardware conditions.

Contribution

It provides a rigorous analysis showing the failure of Shor's algorithm under noisy conditions, establishing noise thresholds beyond which factoring is impossible.

Findings

Shor's algorithm does not factor integers with noise above a small threshold.

The failure probability approaches 1 for random prime pairs under noise.

Noise levels critically impact the effectiveness of quantum factoring algorithms.

Abstract

We consider Shor's quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form $pq$ when the noise exceeds a vanishingly small level in terms of $n$ -- the number of bits of the integer to be factored, where $p$ and $q$ are from a well-defined set of primes of positive density. We further prove that with probability $1 - o(1)$ over random prime pairs $(p,q)$, Shor's factoring algorithm does not factor numbers of the form $pq$, with the same level of random noise present.