Computability Theory of Closed Timelike Curves

We ask, and answer, the question of what’s computable by Turing machines equipped with time travel into the past: that is, closed timelike curves or CTCs (with no bound on their size). We focus on a model for CTCs due to Deutsch, which imposes a probabilistic consistency condition to avoid grandfather paradoxes. Our main result is that computers with CTCs can solve exactly the problems that are Turing-reducible to the halting problem, and that this is true whether we consider classical or quantum computers. Previous work, by Aaronson and Watrous, studied CTC computers with a polynomial size restriction, and showed that they solve exactly the problems in PSPACE, again in both the classical and quantum cases. Compared to the complexity setting, the main novelty of the computability setting is that not all CTCs have fixed-points, even probabilistically. Despite this, we show that the CTCs that do have fixed-points suffice to solve the halting problem, by considering fixed-point distributions involving infinite geometric series. The tricky part is to show that even quantum computers with CTCs can be simulated using a Halt oracle. For that, we need the Riesz representation theorem from functional analysis, among other tools. We also study an alternative model of CTCs, due to Lloyd et al., which uses postselection to “simulate” a consistency condition, and which yields BPPpath in the classical case or PP in the quantum case when subject to a polynomial size restriction. With no size limit, we show that postselected CTCs yield only the computable languages if we impose a certain finiteness condition, or all languages nonadaptively reducible to the halting problem if we don’t.