We consider computation in the presence of closed timelike curves (CTCs), as proposed by Deutsch. We focus on the case in which the CTCs carry classical bits (as opposed to qubits). Previously, Aaronson and Watrous showed that computation with polynomially many CTC bits is equivalent in power to PSPACE. On the other hand, Say and Yakaryılmaz showed that computation with just 1 classical CTC bit gives the power of "postselection", thereby upgrading classical randomized computation (BPP) to the

more »

... mplexity class BPP path and standard quantum computation (BQP) to the complexity class PP. It is natural to ask whether increasing the number of CTC bits from 1 to 2 (or 3, 4, etc.) leads to increased computational power. We show that the answer is no: randomized computation with logarithmically many CTC bits (i.e., polynomially many CTC states) is equivalent to BPP path . (Similarly, quantum computation augmented with logarithmically many classical CTC bits is equivalent to PP.) Spoilsports with no interest in time travel may view our results as concerning the robustness of the class BPP path and the computational complexity of sampling from an implicitly defined Markov chain. On time travel We begin with a discussion of time travel. Readers not interested in this concept may skip directly to Section 2, wherein we define the problem under consideration in a purely complexity-theoretic manner, with no reference to time travel. Kurt Gödel [Göd49] was the first to point out that Einstein's theory of general relativity is consistent with the existence of closed timelike curves (CTCs), raising the theoretical possibility of time travel. Any model of time travel must deal with the "Grandfather Paradox", wherein a trip to the past causes a chain of events that leads to a future in which that very trip does not take place. Assume that a time-traveler changes the state of the universe at the earlier end t 0 of a time loop from state s to some different state s . Then just what is the state of the universe at time t 0 : is it s or s ? Seeing a logical inconsistency in this scenario, most thinkers of earlier generations concluded that time travel to the past must be impossible. There is, however, a way out. In an influential paper [FMN + 90], Friedman et al. suggested Nature might allow CTCs as long as they do not "change the past", an idea that has come to be known as the Novikov self-consistency principle. The main two rivaling models of time travel -the "Deutschian model" (which we study in this work), and the "postselected CTC model" from [LMGP + 10] (which is mentioned in Section 4.2) -both conform to the Novikov self-consistency principle.