Computability Theory of Closed Timelike Curves

Abstract

We study the question of what is computable by Turing machines equipped with time travel into the past; i.e., with Deutschian closed timelike curves (CTCs) having no bound on their width or length. An alternative viewpoint is that we study the complexity of finding approximate fixed points of computable Markov chains and quantum channels of countably infinite dimension. Our main result is that the complexity of these problems is precisely $\Delta_2$, the class of languages Turing-reducible to the Halting problem. Establishing this as an upper bound for qubit-carrying CTCs requires recently developed results in the theory of quantum Markov maps.