Billiard ball CA model running in Golly. (BBM.rle)
Konrad Zuse's "Rechnerraum" (calculating space) (1969), is a seminal work likening the operations of the cosmos to a finite cellular automaton.
Edward Fredkin's work on reversible computing (e.g. Fredkin gates, block automata) led to a concept of digital physics (later digital philosophy) http://www.digitalphilosophy.org.
Discussions with Fredkin influenced Feynman's ideas about Simulating Physics with Computers (1989), where he suggested that quantum processes, or at least stochastic computing, is necessary to simulate physics in finite space without recourse to infinite computational complexity.
John Wheeler summarized his take on a computational universe under a catchy name, "it from bit", in his 1989 essay. Certain conclusions follow, most striking to me is irrational numbers as a "convenient myth".
In a 2012 article, Seth Lloyd reviewed the concept of the computational universe and its history, agreeing with Feynman's earlier assertions that while fully computing physics (including quantum) is intractable with a deterministic, classical computer, the universe as we observe it is consistent with a model as some sort of quantum cellular automata (2012).
An alternative view: hidden variables could explain seemingly random quantum effects, as asserted by Jürgen Schmidhuber (2006) in an appeal to parsimony.