Abstract
We introduce quantum alternation as a generalization of quantum nondeterminism. We define q-alternating Turing machine (qATM) by augmenting alternating Turing machine with constant-size quantum memory. We show that one-way constant-space qATMs (1AQFAs) are Turing equivalent. Then, we introduce strong version of qATM by requiring to halt in every computation path and we show that strong qATMs can simulate deterministic spacewith exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one level. We also focus on realtime versions of 1AQFAs (rtAQFAs) and obtain many results: rtAQFAs can recognize a PSPACE-complete problem; they cannot be simulated by sublinear deterministic Turing machines; for any level of polynomial hierarchy, say k, there exists a complete language that can be recognized by rtAFAs with only (k +1) alternations; and polynomial hierarchy lies in its log-space q-alternation counterpart.
Similar content being viewed by others
References
S. Aaronson, “Quantum computing, postselection, and probabilistic polynomial-time,” Proceedings of the Royal Society A 461 (2063), 3473–3482 (2005).
A. Ambainis and A. Yakaryilmaz, Automata and Quantum Computing. https://arxiv.org/pdf/1507.01988v1.pdf.
S. Arora and B. Barak, Computational Complexity: A Modern Approach (Cambridge University Press, 2009).
L. Babai, “Trading group theory for randomness,” in Proceedings of the 17th Annual ACMSymposium on Theory of Computing (ACM, New York, 1985), pp. 421–429.
A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer, “Alternation,” Journal of the ACM 28 (1), 114–133 (1981).
A. K. Chandra and L. J. Stockmeyer, “Alternation,” in Proceedings of the 17th IEEE Symposium on Foundations of Computer Science (ACM, New York, 1976), pp. 98–108.
A. Condon. Computational Models of Games (MIT Press, 1989).
H.G. Demirci, M. Hirvensalo, K. Reinhardt, A. C. C. Say, and A. Yakaryilmaz, in books@ocg.at, Vol. 304: Proceedings of 6th Workshop on Non-Classical Models of Automata and Applications (NCMA) (Österreichische Computer Gesellschaft,Wien, 2014), pp. 101–114. https://arxiv.org/pdf/1407.0334v1.pdf.
J. Fearnley, Private communications, 2013 and 2014.
J. Fearnley and M. Jurdzinski, “Reachability in two-clock timed automata is PSPACE-complete,” Automata, Languages, and Programming (Springer, Berlin, 2013), pp. 212–223.
S. Fenner, F. Green, S. Homer, and R. Puim, in Sixth Italian Conference on Theoretical Computer Science (World-Scientific, Singapore, 1998), pp. 241–252.
S. Goldwasser, S. Micali, and C. Rackoff, “The knowledge complexity of interactive proof systems,” SIAM Journal on Computing 18 (1), 186–208 (1989).
D. C. Kozen, in FOCS’76: Proceedings of the 17th IEEE Symposium on Foundations of Computer Science (IEEE, New York, 1976), pp. 89–97.
M. Nasu and N. Honda, “A context-free language which is not acceptable by a probabilistic automaton,” Information and Control 18 (3), 233–236 (1971).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
C. H. Papadimitriou, “Games against nature,” Journal of Computer and System Sciences 31 (2), 288–301 (1985).
A. Paz, Introduction to Probabilistic Automata (Academic Press, New York, 1971).
J.H. Reif, “The complexity of two-player games of incomplete information,” Journal of Computer and System Sciences 29 (2), 274–301 (1984).
W. J. Savitch, “Relationships between nondeterministic and deterministic tape complexities,” Journal of Computer and System Sciences 4 (2), 177–192 (1970).
A. C. C. Say and A. Yakaryilmaz, in Computing with New Resources: Essays Dedicated to Jozef Gruska on the Occasion of His 80th Birthday, Ed. by C. S. Calude, R. Freivalds, and I. Kazuo (Springer,New York, 2014), pp. 208–222.
D. van Melkebeek and T. Watson, “Time-space efficient simulations of quantum computations,” Theory of Computing 8 (1), 1–51 (2012).
J. Watrous, Space-bounded quantum computation, PhD Thesis (University of Wisconsin, Madison, 1998).
J. Watrous, “Space-bounded quantum complexity,” Journal of Computer and System Sciences 59 (2), 281–326 (1999).
J. Watrous, “On the complexity of simulating space-bounded quantum computations,” Computational Complexity 12 (1-2), 48–84 (2003).
A. Yakaryilmaz, Turing-Equivalent Automata Using a Fixed-Size Quantum Memory. https://arxiv.org/pdf/1205.5395v1.pdf.
A. Yakaryilmaz, in Computer Science—Theory and Applications, Ed. by A. A. Bulatov and A. M. Shur (Springer, New York, 2013), pp. 366–377.
A. Yakaryilmaz, in The Proceedings of Workshop on Quantum and Classical Complexity (Univeristy of Latvia Press, Riga, 2013), pp. 45–60; Electronic Colloquium on Computational Complexity, TR12-130 (2012).
A. Yakaryilmaz, in Computer Science—Theory and Applications, Ed. by A. A. Bulatov and A. M. Shur (Springer, New York, 2013), pp. 334–346.
A. Yakaryilmaz and A. C. C. Say, “Languages recognized by nondeterministic quantum finite automata,” Quantum Information and Computation 10 (9&10), 747–770 (2010).
A. Yakaryilmaz and A. C. C. Say, “Unbounded-error quantum computation with small space bounds,” Information and Computation 279 (6), 873–892 (2011).
A. Yakaryilmaz and A. C. C. Say, “Proving the power of postselection,” Fundamenta Informaticae 123 (1), 107–134 (2013).
A. Yakaryilmaz, A. C. C. Say, and H. G. Demirci, “Debates with small transparent quantum verifiers,” International Journal of Foundations of Computer Science 27 (2), 283–300 (2016).
T. Yamakami and A. C.-C. Yao, “NQPℂ = co-C=P,” Information Processing Letters 71 (2), 63–69 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by F. M. Ablayev
Rights and permissions
About this article
Cite this article
Yakaryılmaz, A. Quantum alternation. Lobachevskii J Math 37, 637–649 (2016). https://doi.org/10.1134/S1995080216060196
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080216060196