(1) quantum mechanics means that there is not just one state/evolution of the universe. Every possible state/evolution has to be taken into account. Your model is not three-dimensional. It is (NF * NP)-dimensional. NF is the number of fields. NP is the the number of points in space time. So, you want 10 space-time points in a length direction. The universe is four-dimensional so you actually have 10000 space-time points. Now your state space is (10000 * NF)-dimensional. Good luck with that. In fact people try to do such things. I.e., lattice quantum field theory but it is tough.
(2) I am not really sure what the state of the art is but there are problems even with something simple like putting a spin 1/2 particle on a lattice. https://en.wikipedia.org/wiki/Fermion_doubling
(3) Renormalization. If you fancy getting more accuracy by making your lattice spacing smaller, various constants tend to infinity. The physically interesting stuff is the finite part of that. Calculations get progressively less accurate.
To go down this rabbit hole, the deeper question is about the vector in Hilbert space that represents the state of the universe. Is it infinite dimensional?
Yes, but that is not saying very much. Just one single harmonic oscillator already has a state space that is an infinitely dimensional Hilbert space. It is L^2. Now make a tensor product of NF * NP of these already infinitely dimensional Hilbert spaces defined above to get quite a bit more infinite.
(1) quantum mechanics means that there is not just one state/evolution of the universe. Every possible state/evolution has to be taken into account. Your model is not three-dimensional. It is (NF * NP)-dimensional. NF is the number of fields. NP is the the number of points in space time. So, you want 10 space-time points in a length direction. The universe is four-dimensional so you actually have 10000 space-time points. Now your state space is (10000 * NF)-dimensional. Good luck with that. In fact people try to do such things. I.e., lattice quantum field theory but it is tough.
(2) I am not really sure what the state of the art is but there are problems even with something simple like putting a spin 1/2 particle on a lattice. https://en.wikipedia.org/wiki/Fermion_doubling
(3) Renormalization. If you fancy getting more accuracy by making your lattice spacing smaller, various constants tend to infinity. The physically interesting stuff is the finite part of that. Calculations get progressively less accurate.