Computing the N-th term of a q-holonomic sequence

In 1977, Strassen invented a famous baby-step / giant-step algorithm that computes the factorial N! in arithmetic complexity quasi-linear in [EQUATION]. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the N-th term of any holonomic sequence in the same arithmetic complexity. We design q-analogues of these algorithms. We first extend Strassen's algorithm to the computation of the q-factorial of N, then Chudnovskys' algorithm to the computation of the N-th term of any q-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in [EQUATION]. We describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear q-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.