2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/91044We extend Shi's 2002 quantum lower bound for collision in $r$-to-one functions with $n$ inputs. Shi's bound of $Ω((n/r)^{1/3})$ is tight, but his proof applies only in the case where the range has size at least $3n/2$. We give a modified version of Shi's argument which removes this restriction.6 pages Extension of work in quant-ph/0112086Quantum Physicstext