David Tong and Lubos Motl have argued that our universe can't possibly be a digital computer simulation because chiral gauge theories can't be discretized, and the Standard Model is a chiral gauge theory. Certainly, you can't regulate them on a lattice. However, that doesn't mean they're not limit computable. There are only two alternatives. Either chiral gauge theories are uncomputable (extremely unlikely), or they can be simulated on a digital computer. How do you simulate a chiral gauge theory on a digital computer? Attempts by Erich Poppitz have fallen a bit short of the goal.
Overlap fermion approach may be the answer (I think for U(1) gauge symmetry only). Ounce a theory is defined on a lattice, it can be simulated by a computer that we already have. Here is a review on overlap fermion approach:
Tata lectures on overlap fermions arXiv:1103.4588
Overlap formalism deals with the construction of chiral gauge theories on the lattice. These set of lectures provide a pedagogical introduction to the subject with emphasis on chiral anomalies and gauge field topology. Subtleties associated with the generating functional for gauge theories coupled to chiral fermions are discussed.
==== A new result ===
One can simulate any anomaly-free chiral gauge theories on a computer by simply putting it on lattice and turning on a proper direct fermion-fermion interaction. See my new papers http://arxiv.org/abs/1305.1045 and http://arxiv.org/abs/1303.1803
The paper http://arxiv.org/abs/1305.1045 was rejected by PRL (see the referee's comments and my reply http://bbs.sciencenet.cn/home.php?mod=space&uid=1116346&do=blog&id=736247 ). It is now published in CPL.
The papers claim to solve a long standing problem in lattice gauge theory: the chiral fermion problem. So far they fails to attract any attention from lattice gauge community ( http://scholar.google.com/scholar?oi=bibs&hl=en&cites=6976715772443557915 ).
See also this paper: arxiv-1307.7480: Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons. This paper follows Prof. Wen's general thinking and provide a proof between the following two conditions:
"Topological Boundary (Gapping) Conditions"
is equivalent to
"t' Hooft anomaly matching conditions"
The proof is given for the case of the a theory with a U(1) symmetry and in 1+1D.
Using this equivalent relation, one can design the very constrained specific boundary gapping terms to open the mass gap of the mirror sectors.
The untouched sector in principle can provide a lattice chiral fermion model. (or, for the next step, chiral gauge theory in 1+1D.)