Shor算法成功概率的渐近消失
Asymptotic Vanishing of the Success Probability in Shor's Algorithm
AI总结：
Shor算法的成功概率在渐近情况下趋于零，表明其‘预期多项式时间’仅在局部有效，且量子因数分解的可扩展性受限。
摘要：
Shor的因数分解算法保证对于任何固定的模数N=pq（p和q为不同的素数），成功概率至少为一半。我们证明这一保证不适用于渐近情形。当N趋于无穷时，乘法群Omega_N=(Z/NZ)^x形成一个非紧的概率空间族，与成功基相关的概率权重，比例于p(成功|a',N) p(a'|N)，衰减为1/phi(N)。均匀测度族{mu_N}因此不具有弱极限，意味着存在渐近的ergodic性损失。在N<=10^6的蒙特卡罗模拟确认了这一衰减和静态成功概率的不存在。这些结果表明，顺序查找中的“预期多项式时间”仅在局部定义：一旦算术域扩展，就不存在全局期望。渐近成功概率的消失解释了大N情况下Shor算法实现的实证缺失，并设定了量子因数分解可扩展性的基本限制。
英文摘要：
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.
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