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Revision #1 to TR14-091 | 22nd July 2014 13:53

One time-traveling bit is as good as logarithmically many

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Revision #1
Authors: Ryan O'Donnell, A. C. Cem Say
Accepted on: 22nd July 2014 13:53
Downloads: 932
Keywords: 


Abstract:

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 Yakary?lmaz showed that computation with just 1 classical CTC bit gives the power of “postselection”, thereby upgrading classical randomized computation (BPP) to the complexity 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.



Changes to previous version:

Changed the spelling of "traveling" in ECCC's version of the title.


Paper:

TR14-091 | 22nd July 2014 13:48

One time-travelling bit is as good as logarithmically many





TR14-091
Authors: Ryan O'Donnell, A. C. Cem Say
Publication: 22nd July 2014 13:48
Downloads: 989
Keywords: 


Abstract:

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 Yakary?lmaz showed that computation with just 1 classical CTC bit gives the power of “postselection”, thereby upgrading classical randomized computation (BPP) to the complexity 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.



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