On the size of one-way quantum finite automata with periodic behaviors

Mereghetti, Carlo; Palano, Beatrice

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 3, pp. 277-291.

We show that, for any stochastic event $p$ of period $n$, there exists a measure-once one-way quantum finite automaton (1qfa) with at most $2\sqrt{6n}+25$ states inducing the event $ap+b$, for constants $a>0$, $b\ge 0$, satisfying $a+b\le 1$. This fact is proved by designing an algorithm which constructs the desired 1qfa in polynomial time. As a consequence, we get that any periodic language of period $n$ can be accepted with isolated cut point by a 1qfa with no more than $2\sqrt{6n}+26$ states. Our results give added evidence of the strength of measure-once 1qfa’s with respect to classical automata.

Classification : 68Q10, 68Q19, 68Q45
Mots clés : quantum finite automata, periodic events and languages

[1] A. Aho, J. Hopcroft and J. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1974). | MR | Zbl

[2] A. Ambainis, A. Ķikusts and M. Valdats, On the Class of Languages Recognizable by 1-way Quantum Finite Automata, in Proc. 18th Annual Symposium on Theoretical Aspects of Computer Science. Springer, Lecture Notes in Comput. Sci. 2010 (2001) 305-316. | MR | Zbl

[3] A. Ambainis and J. Watrous, Two-way Finite Automata with Quantum and Classical States. Technical Report (1999) quant-ph/9911009. | Zbl

[4] A. Ambainis and R. Freivalds, 1-way Quantum Finite Automata: Strengths, Weaknesses and Generalizations, in Proc. 39th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press (1998) 332-342.

[5] A. Brodsky and N. Pippenger, Characterizations of 1-Way Quantum Finite Automata, Technical Report. Department of Computer Science, University of British Columbia, TR-99-03 (revised). | Zbl

[6] C. Colbourn and A. Ling, Quorums from Difference Covers. Inform. Process. Lett. 75 (2000) 9-12. | MR

[7] L. Grover, A Fast Quantum Mechanical Algorithm for Database Search, in Proc. 28th ACM Symposium on Theory of Computing (1996) 212-219. | MR | Zbl

[8] J. Gruska, Quantum Computing. McGraw-Hill (1999). | MR

[9] J. Gruska, Descriptional complexity issues in quantum computing. J. Autom. Lang. Comb. 5 (2000) 191-218. | MR | Zbl

[10] T. Jiang, E. Mcdowell and B. Ravikumar, The Structure and Complexity of Minimal nfa's over a Unary Alphabet. Int. J. Found. Comput. Sci. 2 (1991) 163-182. | Zbl

[11] A. Ķikusts, A Small 1-way Quantum Finite Automaton. Technical Report (1998) quant-ph/9810065.

[12] M. Kohn, Practical Numerical Methods: Algorithms and Programs. The Macmillan Company (1987). | MR | Zbl

[13] A. Kondacs and J. Watrous, On the Power of Quantum Finite State Automata, in Proc. 38th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press (1997) 66-75.

[14] M. Marcus and H. Minc, Introduction to Linear Algebra. The Macmillan Company (1965). Reprinted by Dover (1988). | MR | Zbl

[15] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities. Prindle, Weber & Schmidt (1964). Reprinted by Dover (1992). | MR | Zbl

[16] C. Mereghetti and B. Palano, Upper Bounds on the Size of One-way Quantum Finite Automata, in Proc. 7th Italian Conference on Theoretical Computer Science. Springer, Lecture Notes in Comput. Sci. 2202 (2001) 123-135. | MR | Zbl

[17] C. Mereghetti, B. Palano and G. Pighizzini, On the Succinctness of Deterministic, Nondeterministic, Probabilistic and Quantum Finite Automata, in Pre-Proc. Descriptional Complexity of Automata, Grammars and Related Structures. Univ. Otto Von Guericke, Magdeburg, Germany (2001) 141-148. RAIRO: Theoret. Informatics Appl. (to appear). | Numdam | MR | Zbl

[18] C. Mereghetti and G. Pighizzini, Two-Way Automata Simulations and Unary Languages. J. Autom. Lang. Comb. 5 (2000) 287-300. | MR | Zbl

[19] C. Moore and J. Crutchfield, Quantum automata and quantum grammars. Theoret. Comput. Sci. 237 (2000) 275-306. | MR | Zbl

[20] J.-E. Pin, On Languages Accepted by finite reversible automata, in Proc. 14th International Colloquium on Automata, Languages and Programming. Springer-Verlag, Lecture Notes in Comput. Sci. 267 (1987) 237-249. | MR | Zbl

[21] P. Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, in Proc. 35th Annual Symposium on Foundations of Computer Science. IEEE Computer Science Press (1994) 124-134. | MR

[22] P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26 (1997) 1484-1509. | MR | Zbl

[23] B. Wichmann, A note on restricted difference bases. J. London Math. Soc. 38 (1963) 465-466. | MR | Zbl