Abstract
We use tools from the algebraic theory of automata to investigate the class of languages recognized by two models of Quantum Finite Automata (QFA): Brodsky and Pippenger’s end-decisive model, and a new QFA model whose definition is motivated by implementations of quantum computers using nucleo-magnetic resonance (NMR). In particular, we are interested in the new model since nucleo-magnetic resonance was used to construct the most powerful physical quantum machine to date. We give a complete characterization of the languages recognized by the new model and by Boolean combinations of the Brodsky-Pippenger model. Our results show a striking similarity in the class of languages recognized by the end-decisive QFAs and the new model, even though these machines are very different on the surface.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambainis, A., Freivalds, R.: 1-way Quantum Finite Automata: Strengths, Weaknesses, and Generalizations. In: Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 332–341 (1998)
Ambainis, A., Ķikusts, A.: Exact Results for Accepting Probabilities of Quantum Automata. Theoretical Computer Science 295, 3–25 (2003)
Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of STOC 2001, pp. 50–59 (2001)
Ambainis, A., Ķikusts, A., Valdats, M.: On the class of Languages Recognized by 1-way Quantum Finite Automata. In: Proceedings of STACS 2001, pp. 75–86 (2001)
Ambainis, A., Nayak, A., Ta-Shma, A., Vazirani, U.: Quantum dense coding and quantum finite automata. Journal of ACM 49, 496–511 (2002)
Beaudry, M., Lemieux, F., Thérien, D.: Finite loops recognize exactly the regular open languages. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, Springer, Heidelberg (1997)
Bertoni, A., Mereghetti, C., Palano, B.: Quantum computing: 1- way quantum finite automata. In: ICM 2002. LNCS, vol. 2730, pp. 1–20 (2003)
Brodsky, A., Pippenger, N.: Characterizations of 1-Way Quantum Finite Automata. SIAM Journal on Computing 31(5), 1456–1478 (2002)
Ciamarra, M.P.: Quantum Reversibility and a New Model of Quantum Automaton. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 376–379. Springer, Heidelberg (2001)
Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, London (1976)
Fuchs, C., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Transactions on Information Theory 45(4), 1216–1227 (1999)
Golovkins, M., Kravtsev, M.: Probabilistic Reversible Automata and Quantum Automata. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 574–583. Springer, Heidelberg (2002)
Gruska, J.: Quantum Computing, p. 160. McGraw-Hill, New York (1999)
Jeavons, P., Cohen, D., Gyssens, M.: Closure Properties of Constraint Satisfaction Problems. JACM 44(4), 527–548 (1997)
Kondacs, A., Watrous, J.: On the power of Quantum Finite State Automata. In: FOCS 1997, pp. 66–75 (1997)
Marshall, A., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic Press, London (1979)
Moore, C., Crutchfield, J.: Quantum Automata and Quantum Grammars. Theoretical Computer Science 237(1-2), 275–306 (2000)
Nayak, A.: Optimal Lower Bounds for Quantum Automata and Random Access Codes. In: Proc. 40th FOCS, pp. 369–377 (1997)
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Pin, J.E.: BG=PG: A success story. NATO Advanced Study Institute Semigroups, Formal Languages and Groups, pp. 33–47 (1995)
Pin, J.E.: On languages accepted by finite reversible automata. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 237–249. Springer, Heidelberg (1987)
Pin, J.E.: A variety theorem without complementation. Russian Mathematics (Izvestija vuzov. Matematika) 39, 80–90 (1995)
Pin, J.E.: Varieties of Formal Languages. North Oxford Academic Publishers, Ltd., London (1986)
Rabin, M.: Probabilistic Automata. Information and Control 6(3), 230–245 (1963)
Simon, I.: Piecewise Testable Events. In: Proc. 2nd GI Conf., pp. 214–222 (1975)
Vandersypen, L., Steffen, M., Breyta, G., Yannoni, C., Sherwood, M., Chuang, I.: Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ambainis, A., Beaudry, M., Golovkins, M., Ķikusts, A., Mercer, M., Thérien, D. (2004). Algebraic Results on Quantum Automata. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-24749-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21236-2
Online ISBN: 978-3-540-24749-4
eBook Packages: Springer Book Archive