Hierarchy and equivalence of multi-letter quantum finite automata

Multi-letter quantum finite automata (QFAs) are a new one-way QFA model proposed recently by Belovs, Rosmanis, and Smotrovs [A. Belovs, A. Rosmanis, J. Smotrovs, Multi-letter reversible and quantum finite automata, in: Proceedings of the 13th International Conference on Developments in Language Theory, DLT'2007, Harrachov, Czech Republic, in: Lecture Notes in Computer Science, vol. 4588, Springer, Berlin, 2007, pp. 60-71], and they showed that multi-letter QFAs can accept with no error some regular languages ((a+b)^*b) that are unacceptable by the one-way QFAs. In this paper, we continue to study multi-letter QFAs. We mainly focus on two issues: (1) we show that (k+1)-letter QFAs are computationally more powerful than k-letter QFAs, that is, (k+1)-letter QFAs can accept some regular languages that are unacceptable by any k-letter QFA. A comparison with the one-way QFAs is made by some examples; (2) we prove that a k"1-letter QFA A"1 and another k"2-letter QFA A"2 are equivalent, if and only if, they are (n"1+n"2)^4+k-1-equivalent, and the time complexity of determining the equivalence of two multi-letter QFAs using this method is O(n^1^2+k^2n^4+kn^8), where n"1 and n"2 are the numbers of states of A"1 and A"2, respectively, and k=max(k"1,k"2). Some other issues are addressed for further consideration.

[1]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[2]  Juraj Hromkovic,et al.  One-way multihead deterministic finite automata , 1983, Acta Informatica.

[3]  D. Deutsch.  Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4]  Cristian S. Calude,et al.  Coins, Quantum Measurements, and Turing's Barrier , 2002, Quantum Inf. Process..

[5]  D. Faddeev,et al.  Computational methods of linear algebra , 1981 .

[6]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[7]  Tero Harju,et al.  Undecidability in ω-Regular Languages , 2006 .

[8]  Mika Hirvensalo.  Quantum computing, Second Edition , 2004, Natural computing series.

[9]  Kazuo Iwama,et al.  Undecidability on quantum finite automata , 1999, STOC '99.

[10]  Daowen Qiu,et al.  Automata theory based on quantum logic: Reversibilities and pushdown automata , 2007, Theor. Comput. Sci..

[11]  Andrew Chi-Chih Yao,et al.  Quantum Circuit Complexity , 1993, FOCS.

[12]  R. Feynman.  Simulating physics with computers , 1999 .

[13]  Mika Hirvensalo,et al.  Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages , 2007, SOFSEM.

[14]  Karel Culik,et al.  On the Equivalence Problem for Deterministic Multitape Automata and Transducers , 1989, STACS.

[15]  James P. Crutchfield,et al.  Quantum automata and quantum grammars , 2000, Theor. Comput. Sci..

[16]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[17]  Ansis Rosmanis,et al.  Multi-letter Reversible and Quantum Finite Automata , 2007, Developments in Language Theory.

[18]  Alex Brodsky,et al.  Characterizations of 1-Way Quantum Finite Automata , 2002, SIAM J. Comput..

[19]  Alberto Bertoni,et al.  Quantum Computing: 1-Way Quantum Automata , 2003, Developments in Language Theory.

[20]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[21]  P. Benioff.  The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines , 1980 .

[22]  Wen-Guey Tzeng,et al.  A Polynomial-Time Algorithm for the Equivalence of Probabilistic Automata , 1992, SIAM J. Comput..

[23]  Alex Kane,et al.  Coins , 1984 .

[24]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[25]  Daowen Qiu,et al.  Determining the equivalence for one-way quantum finite automata , 2007, Theor. Comput. Sci..

[26]  John Watrous,et al.  On the power of quantum finite state automata , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[27]  Tero Harju,et al.  Decidability of the multiplicity equivalence of multitape finite automata , 1990, STOC '90.

[28]  Arto Salomaa,et al.  Computation and Automata , 1984 .

[29]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[30]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[31]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[32]  Daowen Qiu,et al.  Characterization of Sequential Quantum Machines , 2002 .

[33]  Cristian S. Calude,et al.  Reflections on quantum computing , 2000, Complex..

[34]  Mariëlle Stoelinga,et al.  An Introduction to Probabilistic Automata , 2002, Bull. EATCS.

[35]  Andris Ambainis,et al.  1-way quantum finite automata: strengths, weaknesses and generalizations , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[36]  Andris Ambainis,et al.  On the Class of Languages Recognizable by 1-Way Quantum Finite Automata , 2001, STACS.

[37]  Azaria Paz,et al.  Introduction to probabilistic automata (Computer science and applied mathematics) , 1971 .

[38]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[39]  Shou-Feng Wang,et al.  𝒫𝒮-regular languages , 2011, Int. J. Comput. Math..

[40]  Daowen Qiu,et al.  An overview of quantum computation models: quantum automata , 2008, Frontiers of Computer Science in China.

[41]  Daowen Qiu,et al.  Determination of equivalence between quantum sequential machines , 2006, Theor. Comput. Sci..