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On the state complexity of semi-quantum finiteautomata⋆⋆⋆

Published online by Cambridge University Press:  17 April 2014

Shenggen Zheng ,
Jozef Gruska  and
Daowen Qiu
Shenggen Zheng
Affiliation:
Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic.. [email protected]; [email protected]
Jozef Gruska
Affiliation:
Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic.. [email protected]; [email protected]
Daowen Qiu
Affiliation:
Department of Computer Science, Sun Yat-sen University, Guangzhou 510006, P.R. China.; [email protected]
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Abstract

Some of the most interesting and important results concerning quantum finite automata arethose showing that they can recognize certain languages with (much) less resources thancorresponding classical finite automata. This paper shows three results of such a typethat are stronger in some sense than other ones because (a) they deal with models ofquantum finite automata with very little quantumness (so-called semi-quantum one- andtwo-way finite automata); (b) differences, even comparing with probabilistic classicalautomata, are bigger than expected; (c) a trade-off between the number of classical andquantum basis states needed is demonstrated in one case and (d) languages (or the promiseproblem) used to show main results are very simple and often explored ones in automatatheory or in communication complexity, with seemingly little structure that could beutilized.

Keywords

Quantum computingquantum finite automatasemi-quantum finite automatastate complexity
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 48 , Issue 2 , April 2014 , pp. 187 - 207
Copyright
© EDP Sciences 2014

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References

Ambainis, A., Nayak, A., Ta-Shma, A. and Vazirani, U., Dense quantum coding and quantum automata. J. ACM 49 (2002) 496–511. Google Scholar
Ambainis, A., The complexity of probabilistic versus deterministic finite automata, Proc. of the International Symposium on Algorithms and Computation (ISAAC96). Lect. Notes Comput. Sci. 1178 (1996) 233–239. Google Scholar
Ambainis, A. and Watrous, J., Two-way finite automata with quantum and classical states. Theoret. Comput. Sci. 287 (2002) 299–311. Google Scholar
A. Ambainis and R. Freivalds, One-way quantum finite automata: strengths, weaknesses and generalizations, in: Proc. of the 39th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press, Palo Alfo, California, USA (1998) 332–341.
Ambainis, A. and Nahimovs, N., Improved constructions of quantum automata. Theoret. Comput. Sci. 410 (2009) 1916–1922. Google Scholar
Ambainis, A. and Yakaryilmaz, A., Superiority of exact quantum automata for promise problems. Inform. Process. Lett. 112 (2012) 289–291. Google Scholar
Bertoni, A., Mereghetti, C. and Palano, B., Small size quantum automata recognizing some regular languages. Theoret. Comput. Sci. 340 (2005) 394–407. Google Scholar
Bertoni, A., Mereghetti, C. and Palano, b.. Some formal tools for analyzing quantum automata. Theoret. Comput. Sci. 356 (2006) 14–25. Google Scholar
Birget, J.C., State-complexity of finite-state devices, state compressibility and incompressibility. Math. Systems Theory 26 (1993) 237–269. Google Scholar
Brodsky, A. and Pippenger, N., Characterizations of 1-way quantum finite automata. SIAM J. Comput. 31 (2002) 1456–1478. Google Scholar
H. Buhrman, R. Cleve and A. Wigderson, Quantum vs. classical communication and computation. Proc. of 30th Annual ACM Symposium Theory Computing (1998) 63–68.
Buhrman, H., Cleve, R., Massar, S. and de Wolf, R., Nonlocality and Communication Complexity. Rev. Mod. Phys. 82 (2010) 665–698 Google Scholar
Chrobak, M., Finite Automata and Unary Languages. Theoret. Comput. Sci. 47 (1986) 149–158. Google Scholar
Dwork, C. and Stockmeyer, L., A time-complexity gap for two-way probabilistic finite state automata. SIAM J. Comput. 19 (1990) 1011–1023. Google Scholar
Freivalds, R., On the growth of the number of states in result of determinization of probabilistic finite automata. Automatic Control Comput. Sci. 3 (1982) 39–42. Google Scholar
Freivalds, R., Non-constructive methods for finite probabilistic automata. Int. J. Found. Comput. Sci. 19 (2008) 565–580. Google Scholar
Freivalds, R., Ozols, M. and Mancinska, L., Improved constructions of mixed state quantum automata. Theoret. Comput. Sci. 410 (2009) 1923–1931. Google Scholar
W. Feller, An Introduction to Probability Theory and its Applications, vol. I. Wiley, New York, 1967.
J. Gruska, Quantum Computing. McGraw-Hill, London (1999).
Gruska, J., Descriptional complexity issues in quantum computing. J. Automata Languages Combinatorics 5 (2000) 191–218. Google Scholar
J. Gruska, D.W. Qiu, and S.G. Zheng, Communication complexity of promise problems and their applications to finite automata, arXiv:1309.7739 (2013).
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation. Addision-Wesley, New York, 1979.
G.A. Kiraz, Compressed Storage of Sparse Finite-State Transducers, Proc. of CIAA, vol. 2214 of Lect. Notes Comput. Sci. (2001) 109–121.
H. Klauck, On quantum and probabilistic communication: Las Vegas and one-way protocols. Proc. of the 32th STOC (2000) 644–651.
Kushilevitz, E., Communication Complexity. Adv. Comput. 44 (1997) 331–360. Google Scholar
A. Kondacs and J. Watrous, On the power of quantum finite state automata, in Proc. of 38th IEEE Annal. Sympos. Found. of Comput. Sci. (1997) 66–75.
F. Le Gall, Exponential separation of quantum and classical online space complexity, in Proc. of SPAA’06 (2006) 67–73.
Liu, G., Martin-Vide, C., Salomaa, A. and Yu, S., State complexity of basic operations combined with reversal, Inf. Comput. 206 (2008) 1178–186. Google Scholar
Mereghetti, C. and Pighizzini, G., Two-way automata simulations and unary languages. J. Automata, Languages and Combinatorics 5 (2000) 287–300. Google Scholar
Mereghetti, C., Palano, B. and Pighizzini, G., Note on the Succinctness of Deterministic, Nondeterministic, Probabilistic and Quantum Finite Automata. RAIRO: ITA 35 (2001) 477–490. Google Scholar
Mereghetti, C. and Palano, B., On the size of one-way quantum finite automata with periodic behaviors. RAIRO: ITA 36 (2002) 277–291. Google Scholar
Milani, M. and Pighizzini, G., Tight bounds on the simulation of unary probabilistic automata by deterministic automata, J. Automata, Languages and Combinatorics 6 (2001) 481–492. Google Scholar
Mateusa, P., Qiu, D.W. and Li, L.Z., On the complexity of minimizing probabilistic and quantum automata. Inform. Comput. 218 (2012) 36–53. Google Scholar
Moore, C. and Crutchfield, J. P., Quantum automata and quantum grammars. Theoret. Comput. Sci. 237 (2000) 275–306. Google Scholar
A. Paz, Introduction to Probabilistic Automata. Academic Press, New York (1971).
D. W. Qiu, Some Observations on Two-Way Finite Automata with Quantum and Classical States, ICIC 2008. In vol. 5226 of Lect. Notes Comput. Sci. (2008) 1–8.
Rabin, M. O. and Scott, D., Finite automata and their decision problems. IBM J. Research Devel. 3 (1959) 115–125. Google Scholar
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000.
D. W. Qiu, L.Z. Li, P. Mateus and J. Gruska, Quantum finite automata, CRC Handbook of Finite State Based Models and Applications. Edited by J.C. Wang. CRC Press (2012) 113–144.
A. Salomaa, On the reducibility of events represented in automata. In Annales Academiae Scientiarum Fennicae. Volume Series A of I. Math. (1964) 353.
Yu, S., Zhuang, Q. and Salomaa, K.. The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125 (1994) 315–328. Google Scholar
Yu, S., State Complexity: Recent Results and Open Problems. Fundamenta Informaticae 64 (2005) 471–480. Google Scholar
Yakaryilmaz, A. and Cem Say, A.C., Unbounded-error quantum computation with small space bounds, Inform. Comput. 209 (2011) 873–892. Google Scholar
Yakaryilmaz, A. and Cem Say, A.C., Succinctness of two-way probabilistic and quantum finite automata. Discrete Math. Theoret. Comput. Sci. 12 (2010) 19–40. Google Scholar
S.G. Zheng, D.W. Qiu, L.Z. Li and J. Gruska, One-way finite automata with quantum and classical states, vol 7300 of Lect. Notes Comput. Sci. Edited by H. Bordihn, M. Kutrib, and B. Truthe (2012) 273–290.
Zheng, S.G., Qiu, D.W., Gruska, J., Li, L.Z. and Mateus, P., State succinctness of two-way finite automata with quantum and classical states, Theoret. Comput. Sci. 499 (2013) 98–112. Google Scholar
Zheng, S.G., Qiu, D.W. and Li, L.Z., Some languages recognized by two-way finite automata with quantum and classical states. Int. J. Foundation Comput. Sci. 23 (2012) 1117–1129. Google Scholar
S.G. Zheng, J. Gruska and D.W. Qiu. Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata. arXiv:1304.3876 (2013).