Given a divisor q of p-1 of intermediate size, a bounded-error quantum algorithm computes n! mod p in time O~(q^c + sqrt(p/q)), breaking the square-root barrier.
Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier–Manin Operator
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Quantum Algorithms for Modular Factorials
Given a divisor q of p-1 of intermediate size, a bounded-error quantum algorithm computes n! mod p in time O~(q^c + sqrt(p/q)), breaking the square-root barrier.