模阶乘的量子算法

中文摘要 AI 辅助

我们提出了一种有界误差量子算法，给定素数$p$、满足$q\big|(p-1)$的除数$q$以及整数$0<n<p$，该算法能以期望时间$\tilde{O}(q^c+\boldsymbol{\text{sqrt}}(p/q))$计算$n!\bmod p$，其中$c\boldsymbol{\text{ge}}1$为绝对常数。当$p-1$存在大小约为$p^{1/(2c+1)}$的除数$q$时，该算法可达到指数$c/(2c+1)<1/2$。据我们所知，这是首个在该除数承诺下突破模阶乘指数$1/2$壁垒的算法。核心技术是一种量子算法，能以多项式依赖于$q$和$\boldsymbol{\text{log}}p$的复杂度，将相关雅可比和精确重构为紧凑代数形式。我们还将相同渐近界扩展到$0\boldsymbol{\text{le}}n<p^2$范围内$n!\bmod p^2$的计算，当$n=p-1$时，该计算可确定威尔逊商$\frac{(p-1)!+1}{p}\bmod p$。我们推测$q\big|(p-1)$的条件是当前方法的技术限制，而非固有障碍，所有素数都存在统一的量子算法。

英文摘要

We give a bounded-error quantum algorithm that, given a prime $p$, a divisor $q\mid(p-1)$, and an integer $0<n<p$, computes $n!\bmod p$ in expected time $\widetilde{O}(q^c+\sqrt{p/q})$ for some absolute constant $c\ge 1$. When $p-1$ has a divisor of size $q\approx p^{1/(2c+1)}$, this gives the exponent $c/(2c+1)<1/2$. To our knowledge, this is the first algorithm to break the exponent $1/2$ barrier for modular factorials under such a divisor promise. The main technical ingredient is a quantum algorithm that reconstructs the relevant Jacobi sum exactly in compact algebraic form, with polynomial dependence on $q$ and $\log p$. We further extend the same asymptotic bound to the computation of $n!\bmod p^2$, uniformly over $0\le n<p^2$. At $n=p-1$, this determines the Wilson quotient $\frac{(p-1)!+1}{p}\pmod p$. We conjecture that the condition $q\mid(p-1)$ is a technical limitation of the present method rather than an inherent obstruction, and that a uniform quantum algorithm exists for all primes.

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41 pages