There are quantum cellular automata (continuous game of life, which is a type of cellular automata): https://hal.archives-ouvertes.fr/hal-00542373/document
There are continuous cellular automata (continuous game of life which is a type of cellular automata) https://arxiv.org/abs/1111.1567
But I have not seen any quantum continuous cellular automata.
I found a paper (https://arxiv.org/abs/1701.02252) that says that with a Hamiltonian cellular automaton there can be a map between cellular automaton and quantum continuous, but a user in this site, told me that it was not really continuous because time was discrete even in a continuous universe in that model
I also found this paper: https://arxiv.org/abs/1606.08764
There, it is said "We realize constant-space quantum computation by measure-many two-way quantum finite automata" I don't know if that means that this model is a quantum continuous (in space, time...etc) cellular automata/game of life
So do you know of any model that would satisfy what I ask?