One Time-traveling Bit is as Good as Logarithmically Many

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 complexity class BPPpath 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 BPPpath. (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 BPPpath and the computational complexity of sampling from an implicitly defined Markov chain.