Skip to main content
SpringerLink
Log in

Computation with multiple CTCs of fixed length and width

  • Published:
Natural Computing Aims and scope Submit manuscript

Abstract

We examine some variants of computation with closed timelike curves (CTCs), where various restrictions are imposed on the memory of the computer, and the information carrying capacity and range of the CTC. We give full characterizations of the classes of languages decided by polynomial time probabilistic and quantum computers that can send a single classical bit to their own past. We show that, given a time machine with constant negative delay, one can implement CTC-based computations without the need to know about the runtime beforehand. Chaining multiple instances of such fixed-length CTCs, the power of postselection can be endowed to deterministic computers, all languages in \(\mathsf{NP} \cup \mathsf{coNP}\) can be decided with no error in worst-case polynomial time, and all Turing-decidable languages can be decided in constant expected time. We provide proofs of the following facts for weaker models: Augmenting probabilistic computers with a single CTC leads to an improvement in language recognition power. Quantum computers under these restrictions are more powerful than their classical counterparts. Some deterministic models assisted with multiple CTCs are more powerful than those with a single CTC.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Strictly speaking, the amplitudes in (Aaronson and Watrous 2009) are complex numbers whose real and imaginary parts are rationals. We use only rational amplitudes in the definition. This does not change the computational power of the resulting model, and makes some proofs easier.

  2. David Deutsch, personal communication.

  3. Our definition of \(\mathsf{BQP_{CTC_1}}\) is different from the \(\mathsf{BQP_{CTC1}}\) given in (Aaronson and Watrous 2009), since Aaronson and Watrous consider a quantum bit sent through the CTC.

  4. The definition in (Aaronson 2005) allows efficiently computable irrational numbers as transition amplitudes, whereas Q has only rational numbers in its description. Fortunately, \(\mathcal{A}\) can be assumed to have rational amplitudes by (Aaronson 2005), (Adleman et al. 1997), and (Yao 1993).

  5. See (Brun and Wilde 2012) for some examples of the procedure used in the simulation of programs with postselection by P-CTC-assisted circuits.

  6. P-CTCs augmenting classical core models have not been studied before, but our reasoning above applies to that case as well, indicating that such computers can decide precisely the languages in \(\mathsf{BPP_{path}}\) in polynomial time, and with a single CTC bit.

  7. The finite automata under consideration can make a single pass over the input, and do not possess the computational power to measure its length anyway. This is reminiscent of the definition of \(\mathsf{AC_{CTC}^{0}}\) in (Aaronson and Watrous 2009), where one uses a stronger machine (a deterministic TM) to prepare a CTC of the appropriate width for augmenting a weaker model (\(\mathsf{AC^{0}}\)). We examine an alternative to this approach in the next section.

  8. One can obtain a new QFA that is the tensor product of the two given QFAs for this construction.

  9. The proof in (Yakaryılmaz and Say 2011a) is actually more general, and shows that real-time QTMs using space s are more powerful than their probabilistic counterparts for all \(s\in o(\log n)\). Appropriately defined CTC-assisted versions of these two models would then be related in the same way, by Theorem 1.

  10. Note that such a model (Morris et al. 1988) was mentioned briefly in the beginning of Sect. 2.

  11. David Deutsch, personal communication.

  12. Although each coin CTC in Fig. 2 overlaps with the next one, they do not form a “chain” where the bit value is forced to be the same everywhere, as in the decision bit CTCs. The overlap is an artifact of having to use the same fixed-length channel for multiple bits.

References

  • Aaronson S (2005) Quantum computing, postselection, and probabilistic polynomial-time. Proc R Soc A 461(2063):3473–3482

    Article  MathSciNet  MATH  Google Scholar 

  • Aaronson S, Watrous J (2009) Closed timelike curves make quantum and classical computing equivalent. Proc R Soc A 465(2102):631–647

    Article  MathSciNet  MATH  Google Scholar 

  • Adleman LM, DeMarrais J, Huang M-DA (1997) Quantum computability. SIAM J Comput 26(5):1524–1540

    Article  MathSciNet  MATH  Google Scholar 

  • Ambainis A, Watrous J (2002) Two–way finite automata with quantum and classical states. Theoret Comput Sci 287(1):299–311

    Article  MathSciNet  MATH  Google Scholar 

  • Bacon D (2004) Quantum computational complexity in the presence of closed timelike curves. Phys Rev A 70(032309)

  • Brun TA, Wilde MM (2012) Perfect state distinguishability and computational speedups with postselected closed timelike curves. Found Phys 42(3):341–361

    Article  MathSciNet  MATH  Google Scholar 

  • Deutsch D (1991) Quantum mechanics near closed timelike lines. Phys Rev D 44(10):3197–3217

    Article  MathSciNet  Google Scholar 

  • Dwork C, Stockmeyer L (1990) A time complexity gap for two-way probabilistic finite-state automata. SIAM J Comput 19(6):1011–1123

    Article  MathSciNet  MATH  Google Scholar 

  • Freivalds R (1981) Probabilistic two-way machines. In: Proceedings of the International Symposium on Mathematical Foundations of Computer Science, pp 33–45

  • Gray JN, Harrison MA, Ibarra OH (1967) Two-way pushdown automata. Inf Control 11:30–70

    Article  MathSciNet  MATH  Google Scholar 

  • Herzog C (1997) Pushdown automata with bounded nondeterminism and bounded ambiguity. Theoret Comput Sci 181:141–157

    Article  MathSciNet  MATH  Google Scholar 

  • Hoogeboom HJ, Engelfriet J (2004) Pushdown automata. In: Formal languages and applications. (Studies in fuzziness and soft computing, vol 148), chap 6. Springer, New York, pp 117–138

  • Hromkovič J, Schnitger G (2010) On probabilistic pushdown automata. Inf Comput 208(8):982–995

    Article  MATH  Google Scholar 

  • Lloyd S, Maccone L, Garcia-Patron R, Giovannetti V, Shikano Y (2010a) The quantum mechanics of time travel through post-selected teleportation. Technical report, arXiv:1007.2615

  • Lloyd S, Maccone L, Garcia-Patron R, Giovannetti V, Shikano Y, Pirandola S, Rozema LA, Darabi A, Soudagar Y, Shalm LK, Steinberg AM (2010b) Closed timelike curves via post-selection: Theory and experimental demonstration. Technical report, arXiv:1005.2219

  • Morris MS, Thorne KS, Yurtsever U (1988) Wormholes, time machines, and the weak energy condition. Phys Rev Lett 61(13):1446–1449

    Article  Google Scholar 

  • Rabin MO (1963) Probabilistic automata. Inf Control 6:230–243

    Article  Google Scholar 

  • Shepherdson JC (1959) The reduction of two–way automata to one-way automata. IBM J Res Dev 3:198–200

    Article  MathSciNet  Google Scholar 

  • Sipser M (2006) Introduction to the theory of computation, 2nd edn. Thomson Course Technology, Boston

    MATH  Google Scholar 

  • Wotschke D (1978) Nondeterminism and boolean operations in PDA’s. J Comput Syst Sci 16(3):456–461

    Google Scholar 

  • Yakaryılmaz A, Say ACC (2010) Succinctness of two-way probabilistic and quantum finite automata. Discr Math Theoret Comput Sci 12(4):19–40

    Google Scholar 

  • Yakaryılmaz A, Say ACC (2011a) Proving the power of postselection. Technical report, arXiv:1111.3125 (Prepublication version of paper accepted for publication in Fundamenta Informaticae in 2012)

  • Yakaryılmaz A, Say ACC (2011b) Unbounded-error quantum computation with small space bounds. Inf Comput 279(6):873–892

    Article  Google Scholar 

  • Yao ACC (1993) Quantum circuit complexity. In FOCS1993: Proceedings of the Thirty-fourth IEEE Symposium on Foundations of Computer Science, pp 352–361

Download references

Acknowledgements

We thank David Deutsch, Scott Aaronson, Seth Lloyd, Amos Ori, Todd Brun, Taylan Cemgil, and Charles Bennett for their helpful answers and remarks. We are also grateful to all the anonymous reviewers of the past and present versions of this manuscript for their insightful comments. We were partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) with grant 108E142. Yakaryılmaz was also partially supported by the FP7 FET-Open project QCS.

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Boğaziçi University, Bebek, 34342, Istanbul, Turkey

    A. C. Cem Say

  2. Faculty of Computing, University of Latvia, Raiņa bulvāris 19, Riga, 1586, Latvia

    Abuzer Yakaryılmaz

Authors
  1. A. C. Cem Say
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abuzer Yakaryılmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. C. Cem Say.

Additional information

A preliminary version of this work was presented in the 10th International Conference on Unconventional Computation (UC2011).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Say, A.C.C., Yakaryılmaz, A. Computation with multiple CTCs of fixed length and width. Nat Comput 11, 579–594 (2012). https://doi.org/10.1007/s11047-012-9337-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11047-012-9337-6

Keywords

Search

Navigation