Abstract
Quantum automata, as theoretical models of quantum computers, include quantum finite automata (QFA), quantum sequential machines (QSM), quantum pushdown automata (QPDA), quantum Turing machines (QTM), quantum cellular automata (QCA), and the others, for example, automata theory based on quantum logic (orthomodular lattice-valued automata). In this paper, we try to outline a basic progress in the research on these models, focusing on QFA, QSM, QPDA, QTM, and orthomodular lattice-valued automata. Also, other models closely relative to them are mentioned. In particular, based on the existing results in the literature, we finally address a number of problems to be studied in future.
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References
Gruska J. Quantum Computing. London: McGraw-Hill, 1999
Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000
Bennett C. Logical reversibility of computation. IBM Journal of Research and Development, 1973, 17: 525–532
Benioff P. The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. Journal of Statistical Physics, 1980, 22: 563–591
Feynman R P. Simulating physics with computers. International Journal of Theoretical Physics, 1982, 21: 467–488
Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London A, 1985, 400: 97–117
Yao A C. Quantum circuit complexity. In: Proceedings of the 34th IEEE Symposium on Foundations of Computer science. IEEE Computer Society Press, 1993, 352–361
Shor P W. Algorithm for quantum computation: Discrete logarithms and factoring. In: Proceedings of the37th IEEE Annuual Symposium on Foundations of Computer science. IEEE Computer Society Press, 1994, 124–134
Grover L. A fast quantum mechanical algorithms for datdbase search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing. New York: ACM, 1996, 212–219
Hopcroft J E, Ullman J D. Introduction to Automata Theory, Languages, and Computation. New York: Addision-Wesley, 1979
Moore C, Crutchfield J P. Quantum automata and quantum grammars. Theoretical Computer Science, 2000, 237: 275–306. Also see arXiv: quant-ph/9707031, 1997
Kondacs A, Watrous J. On the power of finite state automata. In: Proceedings of the 38th IEEE Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press, 1997, 66–75
Ambainis A, Freivalds R. One-way quantum finite automata: strengths, weaknesses and generalizations. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science. Palo Alfo, California: IEEE Computer Society Press, 1998, 332–341, also see arXiv: quant-ph/9802062, 1998
Nayak A. Optimal lower bounds for quantum automata and random access codes. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science. IEEE Computer Society, 1999, 124–133
Ambainis A, Nayak A, Ta-Shma A, et al. Dense quantum coding and quantum automata. Journal of the ACM 2002, 49(4): 496–511
Ambainis A, Beaudry M, Golovkins M, et al. Algebraic results on quantum automata. Theory of Computing Systems, 2006, 39: 1654–1188
Amano M, Iwama K. Undecidability on quantum finite automata. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing. Atlanta, Georgia: ACM Computer Society Press, 1999, 368–375
Ambainis A, Watrous J. Two-way finite automata with quantum and classical states. Theoretical Computer Science, 2002, 287: 299–311
Bertoni A, Carpentieri M. Analogies and differences between quantum and stochastic automata. Theoretical Computer Science, 2001, 262: 69–81
Bertoni A, Carpentieri M. Regular Languages accepted by quantum automata. Information and Computation, 2001, 165: 174–182
Blondel V D, Jeandel E, Koiran P, et al. Decidable and undecidable problems about quantum automata. SIAM Journal on Computing, 2005, 34(6): 1464–1473
Bertoni A, Mereghetti C, Palano B. Quantum computing: 1-way quantum automata. In: Proceedings of the 9th International Conference on Developments in Language Theory (DLT’2003), Lecture Notes in Computer Science, 2003, 2710: 1–20
Brodsky A, Pippenger N. Characterizations of 1-way quantum finite automata. SIAM Journal on Computing, 2002, 31: 1456–1478, also seequant-ph/9903014, 1999
Ambainis A, Kikusts A, Valdats M. On the class of languages recognizable by 1-way quantum finite automata. In: Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, 2001, 2010: 305–316
Qiu D W. Characterizations of quantum automata. Journal of Software, 2003, 14(1): 9–15, (in Chinese)
Qiu D W, Ying M S. Characterizations of quantum automata. Theoretical Computer Science, 2004, 312: 479–489
Qiu D W. Some observations on two-way finite automata with quantum and classical states. see quant-ph/0701187, 2007
Li L Z, Qiu D W. Determining the equivalence for one-way quantum finite automata. See quant-ph/0703087v2, 2007. Theoretical Computer Science (in press)
Gudder S. Quantum computers. International Journal of Theoretical Physics, 2000, 39: 2151–2177
Qiu D W. Characterization of sequential quantum machines. International Journal of Theoretical Physics, 2002, 41: 811–822
Li L Z, Qiu D W. Determination of equivalence between quantum sequential machines. Theoretical Computer Science, 2006, 358: 65–74
Golovkins M. Quantum pushdown automata. In: Proceedings of the 27th Conference on Current Trends in Theory and Practice of Informatics. Lecture Notes in Computer Science, 2000, 1963: 336–346
Nakanishi M. On the power of one-sided error quantum pushdown automata with classical stack operations. In: Proceedings of the 10th Annual International Computing and Combinatorics Conference (COCOON 2004). Lecture Notes in Computer Science, 2004, 3106: 179–187
Bernstein E, Vazirani U. Quantum complexity theory. SIAM Journal on Computing, 1997, 26(5): 1411–1473
Nishimura H, Ozawa M. Computational complexity of uniform quantum circuit families and quantum Turing machines. Theoretical Computer Science, 2002, 276: 147–181
Watrous J. Space-bounded quantum complexity. Journal of Computer and System Sciences, 1999, 59(2): 281–326
Hirvensalo M. On quantum computation. PhD thesis, Turku Center for Computer Science, 1997
Qiu D W. Simulations of quantum Turing machines by quantum multi-stack machines. In: Computability in Europe (CIE2008), Athens, Greece, June 15–20, 2008, also see arXiv: quant-ph/0501176, 2005
Müller M. Strongly universal quantum Turing machines and invariance of Kolmogorov complexity. IEEE Transactions on Information Theory, 2008, 54(2): 763–780
Watrous J. On one-dimensional cellular automata. In: Proceedings of the 36th IEEE Annual Symposium on Foundations of Computer Science. 1995, 528–537
Dürr C, Santha M. A decision procedure for unitary linear quantum cellular automata. SIAM Journal on Computing, 2002, 31(4): 1076–1089
Ying M S. Automata theory based on quantum logic I. International Journal of Theoretical Physics, 2000, 39: 981–991
Ying M S. Automata theory based on quantum logic II. International Journal of Theoretical Physics, 2000, 39: 2545–2557
Qiu D W. Automata and grammar theory based on quantum logic. Journal of Software, 2003, 14(1): 23–27, (in Chinese)
Qiu D W. Automata theory based on quantum logic: Some characterizations. Information and Compututation, 2004, 190: 179–195
Lu R Q, Zheng H. Lattices of quantum automata. International Journal of Theoretical Physics, 2003, 42: 1425–1449
Cheng W, Wang J. Grammar theory based on quantum logic. International Journal of Theoretical Physics, 2003, 42: 1677–1691
Ying M S. A theory of computation based on quantum logic (I). Theoretical Computer Science, 2005, 344: 134–207
Qiu D W. Automata theory based on quantum logic: reversibilities and pushdown automata. Theoretical Computer Science, 2007, 386: 38–56
Qiu D W. Notes on automata theory based on quantum logic. Science in China (Series F: Information Sciences), 2007, 50(2): 154–169
Rabin M O. Probabilistic automata. Information and Control, 1963, 6: 230–244
Paz A. Introduction to Probabilistic Automata. New York: Academic Press, 1971
Li L Z, Qiu D W. A polynomial-time algorithm for the equivalence between quantum sequential machines. see arXiv: quant-ph/0604085, 2006
Tonder A V. A lambda calculus for quantum computation. SIAM Journal on Computing, 2004, 33(5): 1109–1135
Gudder S. Quantum automata: An overview. International Journal of Theoretical Physics, 1999, 38: 2261–2282
Koshiba T. Polynomial-time algorithms for the equivalence for one-way quantum finite automata. In: Proceedings of the 12th International Symposium on Algorithms and Computation (ISAAC’2001), Christchurch, New Zealand, Lecture Notes in Computer Science, 2001, 2223: 268–278
Ambainis A, Bonner R, Freivalds R, et al. Probabilities to accept languages by quantum finite automata. In: Proceedings of COCOON’99, Lecture Notes in Computer Science, 1999, 1627: 174–183
Ambainisa A, Kikusts A. Exact results for accepting probabilities of quantum automata. Theoretical Computer Science, 2003, 295: 3–25
Kikusts A. A small 1-way quantum finite automaton. see arXiv: quant-ph/9810065, 1998
Ablayev F, Gainutdinova A. On the Lower Bounds for One-Way Quantum Automata. In: Proceedings of 25th International Symposium on Mathematical Foundations of Computer Science (MFCS’2000), Lecture Notes in Computer Science, 2000, 1893: 132–140
Bertoni A, Mereghetti C, Palano B. Lower Bounds on the Size of Quantum Automata Accepting Unary Languages. In: Proceedings of 8th Italian Conference on Theoretical Computer Science (ICTCS’2003), Lecture Notes in Computer Science, 2003, 2841: 86–96
Bertoni A, Mereghetti C, Palano B. Small size quantum automata recognizing some regular languages. Theoretical Computer Science, 2005, 340: 394–407
Mereghetti C, Palano B. Upper Bounds on the Size of One-Way Quantum Finite Automata, In: Proceedings of the 7th Italian Conference on Theoretical Computer Science (ICTCS’2001), Lecture Notes in Computer Science, 2001, 2202: 123–135
Mereghetti C, Palano B. On the size of one-way quantum finite automata with periodic behaviors. Theoretical Informatics and Applications, 2002, 36: 277–291
Mereghetti C, Palano B. Quantum automata for some multiperiodic languages. Theoretical Computer Science, 2007, 387: 177–186
Paschen K. Quantum finite automata using ancilla qubits. University of Karlsruhe, Technical Report, May 2000
Freivalds R. Probabilistic two-way machines. In: Proceedings of the International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, 1981, 188: 33–45
Greenberg A, Weiss A. A lower bound for probabilistic algorithms for finite state machines. Journal of Computer and System Sciences, 1986, 33(1): 88–105
Dwork C, Stockmeyer L. A time-complexity gap for two-way probabilistic finite state automata. SIAM Journal on Computing, 1990, 19: 1011–1023
Kaneps J, Freivalds R. Running time to recognize nonregular languages by 2-way probabilistic automata. In: Proceedings of the 18th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science 1991, 510: 174–185
Dwork C, Stockmeyer L. Finite state verifier I: the power of interaction. Journal of the ACM, 1992, 39(4): 800–828
Birkhoff G, von Neumann J. The logic of quantum mechanics. Annals of Mathematics, 1936, 37: 823–843
Pták P, Pulmannová S. Orthomodular Structures as Quantum Logics. Dordrecht: Kluwer, 1991
Qiu D W. Automata theory based on complete residuated lattice-valued logic. Science in China (Series F: Information Sciences), 2001, 44(6): 419–429
Qiu D W. Automata theory based on complete residuated lattice-valued logic (II). Science in China (Series F: Information Sciences), 2002, 45(6): 442–452
Ledda A, Konig M, Paoli F, et al. MV-algebras and quantum computation. Studia Logica, 2006, 82: 245–270
Deutsch D. Quantum computational networks. Proceedings of the Royal Society of London Series A, 1989, 400: 73–90
Adleman L, DeMarrais J, Huang H. Quantum computability. SIAM Journal on Computing, 1997, 26(5): 1524–1540
Bennett C, Bernstein E, Brassard G, et al. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 1997, 26(5): 1510–1523
Cleve R. An Introduction to Quantum Complexity Theory. arXiv: quantph/9906111v1, 1999
Fortnow L, Rogers J. Complexity limitations on quantum computation. Journal of Computer and System Sciences, 1999, 59(2): 240–252
Fortnow L. One complexity theorists view of quantum computing. Theoretical Computer Science, 2003, 292: 597–610
Simon D. On the power of quantum computation. SIAM Journal on Computing, 1997, 26(5): 1474–1483
Spakowski H, Thakur M, Tripathi R. Quantum and classical complexity classes: Separations, collapses, and closure properties. Information and Computation, 2005, 200(1): 1–34
Yu S. Regular Languages. In: Rozenberg G, Salomaa A, eds. Handbook of Formal Languages. Berlin: Spring-Verlag, 1998, 41–110
Nishimura H, Yamakami T. An application of quantum finite automata to interactive proof systems. In: Proceedings of the 9th International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science. Berlin: Springer, 2004, 3317: 225–236
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Qiu, D., Li, L. An overview of quantum computation models: quantum automata. Front. Comput. Sci. China 2, 193–207 (2008). https://doi.org/10.1007/s11704-008-0022-y
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DOI: https://doi.org/10.1007/s11704-008-0022-y