Is photon interference really random? I know that according to the many worlds interpretation, there is no randomness and rather there is a universal wave function that simulates an observer with a continuously branching timeline. My question is if we assume that the theory simulates only one timeline, can we deduce that the the distribution of points in space-time where a photon is absorbed really is algorithmically random? 
My question is not whether apparent randomness can be explained by the chaotic nature of hidden variables. If the universe follows such a hidden variable theory and the state at the beginning of time is algorithmically random, the fully deterministic chaotic nature will still produce an algorithmically random sequence of observations. My question is whether the distribution of points in space time where photons get absorbed is random, not whether the future can't be predicted from the present and if the universe follows such a theory of hidden variables with an algorithmically random state at the beginning of time, the answer will still be yes. 
Have researchers submitted a photon interference pattern into a computer and run an algorithm to compute whether it has all sorts of properties of randomness? If they did and it computed it not to have all those random properties, then quantum interference is not random. If not, that doesn't necessarily mean it's random. In fact, if it is random, we can never prove it's random. It's easy to prove that for any computable function no matter how rapidly growing, there exists an algorithm that computes within the time of a more rapidly growing function, a number that is algorithmically random within the time of that function.
 A: [After discussion with OP, this does not answer the question as intended]
Quantum Theory, as originally formulated, is a stochastic theory. That is the basic mathematical terms in the theory provide predictions in the form of a probability distribution. No attempt is made to explain the algorithmic origin of these distributions.
As you point out, the many worlds interpretation of quantum theory is presented as deterministic theory. However the special way it turns quantum theory into a deterministic theory result in the theory having no more predictive power than the original quantum theory. (It achieves this by essentially giving an anthropic principle type explanation for the apparent quantum randomness).
As a result your question "Have researchers submitted a photon interference pattern into a computer and run an algorithm to compute whether it has all sorts of properties of randomness?" is as ill founded for many worlds theory as it is for the original stochastically formulated quantum theory.
The photon interference pattern only ever provides probabilistic information in either theory. 
