Space-time symmetric qubit regularization of asymptotically freedom

We explore if space-time symmetric lattice field theory models with a finite Hilbert space per lattice site can reproduce asymptotic freedom. To make things concrete we consider the continuum two dimensional O(4) model which is asymptotically free. Within a class of simple lattice models, we discover a model with a four dimensional local Hilbert space that can reproduce the step scaling function of the continuum model up to correlation lengths of about 50 lattice sites. As expected this requires some fine tuning, without which the correlation lengths reached are usually much smaller. On the other hand, using D-theory ideas we can reach systematically larger correlations, but this naturally increases the Hilbert space per site. We argue that while the D-theory approach increases the local Hilbert space, the complexity of quantum computation is likely to be manageable due to locality in the extra dimension. However, the search for a space-time symmetric lattice model with a finite fixed local Hilbert space that reproduces the step scaling function for arbitrarily large correlation lengths remains ongoing.