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.
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Acknowledgements
This work was finished while the author was attending the International Congress of Basic Science in Beijing. The author wants to express his sincere gratitude to the anonymous referees for their careful reading and comments. He also thanks Al AHO, Dan BONEH, Péter GÁCS, Zvi GALIL, Fred GREEN, Steve HOMER, Gil KALAI, Leonid LEVIN, Dick LIPTON, Ashwin MARAN, Albert MEYER, Ken REGAN, Ron RIVEST, Peter SHOR, Mike SIPSER, Les VALIANT, and Ben YOUNG for insightful comments. And he particularly thanks Eric BACH for inspiring discussions on some of the number theoretic estimates. A similar result can be proven for Shor’s algorithm computing discrete logarithm, and will be reported later.
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Cai, JY. Shor’s algorithm does not factor large integers in the presence of noise. Sci. China Inf. Sci. 67, 173501 (2024). https://doi.org/10.1007/s11432-023-3961-3
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DOI: https://doi.org/10.1007/s11432-023-3961-3