Shor's Algorithm Does Not Factor Large Integers in the Presence of Noise
Jin-Yi Cai
TL;DR
This work analyzes Shor's quantum factoring algorithm under a model of independent random angle perturbations on controlled rotation gates and proves a provable failure regime for factoring $n$-bit numbers of the form $N=pq$ when the noise level satisfies $\epsilon > c\,n^{-1/3}$. By combining a banded variant of Shor's circuit with number-theoretic structure supplied by Fouvry's theorem, the author shows that a random element of $\mathbb{Z}_N^*$ typically has large order, enabling a lattice-based argument to bound the coherence of the noisy quantum Fourier transform. The main result asserts that, for primes from a positive-density set (and, with probability $1-o(1)$, for random primes of a given size), Shor's algorithm fails to factor $N=pq$ with exponentially small success probability when the noise is present, highlighting a fundamental sensitivity of quantum factoring to noise. The appendix extends the conclusion to random primes of length $m$ and discusses broader implications, including connections to the banded Shor approach and the discussion of the Strong Church-Turing thesis.
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.