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.

25 pages; v1 contained an erroneous proof of the main theorem (Theorem 10). A correction is given in Theorem 3.3

References in corpus (4)

• NP-complete Problems and Physical Reality
• The quantum mechanics of time travel through post-selected teleportation
• Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?
• The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

Cited by in corpus (5)

• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Unlimited non-causal correlations and their relation to non-locality
• Computational tameness of classical non-causal models
• The Algebraic View of Computation
• Revisiting Integer Factorization using Closed Timelike Curves