Quantum Algorithms for Modular Factorials

Abstract: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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2607.29453 [quant-ph]

(or arXiv:2607.29453v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.29453 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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