Asymptotic Vanishing of the Success Probability in Shor's Algorithm
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918268088877056 |
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| author | da Cruz, João P. |
| author_facet | da Cruz, João P. |
| contents | Shor's factoring algorithm guarantees a success probability of at least one half for any fixed modulus N = pq with distinct primes p and q. We show that this guarantee does not extend to the asymptotic regime. As N -> infinity, the multiplicative groups Omega_N = (Z/NZ)^x form a non-tight family of probability spaces, and the probability weight associated with successful bases, proportional to p(success | a', N) p(a' | N), decays as 1/phi(N). The ensemble of uniform measures {mu_N} therefore admits no weak limit, implying an asymptotic loss of ergodicity. Monte Carlo simulations up to N <= 10^6 confirm this decay and the absence of a stationary success probability. These results demonstrate that the "expected polynomial time" in order finding is only locally defined: no global expectation exists once the arithmetic domain expands. The asymptotic vanishing of success probability explains the empirical absence of large-N implementations of Shor's algorithm and sets a fundamental limit on the scalability of quantum factoring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Vanishing of the Success Probability in Shor's Algorithm da Cruz, João P. Quantum Physics Mathematical Physics Computational Physics Shor's factoring algorithm guarantees a success probability of at least one half for any fixed modulus N = pq with distinct primes p and q. We show that this guarantee does not extend to the asymptotic regime. As N -> infinity, the multiplicative groups Omega_N = (Z/NZ)^x form a non-tight family of probability spaces, and the probability weight associated with successful bases, proportional to p(success | a', N) p(a' | N), decays as 1/phi(N). The ensemble of uniform measures {mu_N} therefore admits no weak limit, implying an asymptotic loss of ergodicity. Monte Carlo simulations up to N <= 10^6 confirm this decay and the absence of a stationary success probability. These results demonstrate that the "expected polynomial time" in order finding is only locally defined: no global expectation exists once the arithmetic domain expands. The asymptotic vanishing of success probability explains the empirical absence of large-N implementations of Shor's algorithm and sets a fundamental limit on the scalability of quantum factoring. |
| title | Asymptotic Vanishing of the Success Probability in Shor's Algorithm |
| topic | Quantum Physics Mathematical Physics Computational Physics |
| url | https://arxiv.org/abs/2510.06271 |