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

Carlo Mereghetti; Beatrice Palano

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2002)

  • Volume: 36, Issue: 3, page 277-291
  • ISSN: 0988-3754

Abstract

top
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 6 n + 25 states inducing the event a p + b , for constants a > 0 , b ≥ 0 , satisfying a + b ≤ 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 6 n + 26 states. Our results give added evidence of the strength of measure-once 1qfa’s with respect to classical automata.

How to cite

top

Mereghetti, Carlo, and Palano, Beatrice. "On the size of one-way quantum finite automata with periodic behaviors." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 36.3 (2002): 277-291. <http://eudml.org/doc/245040>.

@article{Mereghetti2002,
abstract = {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 $a&gt;0$, $b\!\ge 0$, 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.},
author = {Mereghetti, Carlo, Palano, Beatrice},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {quantum finite automata; periodic events and languages; measure-once one-way quantum finite automaton},
language = {eng},
number = {3},
pages = {277-291},
publisher = {EDP-Sciences},
title = {On the size of one-way quantum finite automata with periodic behaviors},
url = {http://eudml.org/doc/245040},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Mereghetti, Carlo
AU - Palano, Beatrice
TI - On the size of one-way quantum finite automata with periodic behaviors
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 3
SP - 277
EP - 291
AB - 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 $a&gt;0$, $b\!\ge 0$, 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.
LA - eng
KW - quantum finite automata; periodic events and languages; measure-once one-way quantum finite automaton
UR - http://eudml.org/doc/245040
ER -

References

top
  1. [1] A. Aho, J. Hopcroft and J. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1974). Zbl0326.68005MR413592
  2. [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. Zbl0976.68087MR1890780
  3. [3] A. Ambainis and J. Watrous, Two-way Finite Automata with Quantum and Classical States. Technical Report (1999) quant-ph/9911009. Zbl1061.68047
  4. [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. [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). Zbl1051.68062
  6. [6] C. Colbourn and A. Ling, Quorums from Difference Covers. Inform. Process. Lett. 75 (2000) 9-12. MR1774826
  7. [7] L. Grover, A Fast Quantum Mechanical Algorithm for Database Search, in Proc. 28th ACM Symposium on Theory of Computing (1996) 212-219. Zbl0922.68044MR1427516
  8. [8] J. Gruska, Quantum Computing. McGraw-Hill (1999). MR1978991
  9. [9] J. Gruska, Descriptional complexity issues in quantum computing. J. Autom. Lang. Comb. 5 (2000) 191-218. Zbl0965.68021MR1778471
  10. [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. Zbl0746.68040
  11. [11] A. Ķikusts, A Small 1-way Quantum Finite Automaton. Technical Report (1998) quant-ph/9810065. 
  12. [12] M. Kohn, Practical Numerical Methods: Algorithms and Programs. The Macmillan Company (1987). Zbl0665.65002MR892962
  13. [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. [14] M. Marcus and H. Minc, Introduction to Linear Algebra. The Macmillan Company (1965). Reprinted by Dover (1988). Zbl0142.26801MR1019834
  15. [15] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities. Prindle, Weber & Schmidt (1964). Reprinted by Dover (1992). Zbl0126.02404MR1215484
  16. [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. Zbl1042.68066MR1915411
  17. [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). Zbl1010.68068MR1908867
  18. [18] C. Mereghetti and G. Pighizzini, Two-Way Automata Simulations and Unary Languages. J. Autom. Lang. Comb. 5 (2000) 287-300. Zbl0965.68043MR1778478
  19. [19] C. Moore and J. Crutchfield, Quantum automata and quantum grammars. Theoret. Comput. Sci. 237 (2000) 275-306. Zbl0939.68037MR1756213
  20. [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. Zbl0627.68069MR912712
  21. [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. MR1489242
  22. [22] P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26 (1997) 1484-1509. Zbl1005.11065MR1471990
  23. [23] B. Wichmann, A note on restricted difference bases. J. London Math. Soc. 38 (1963) 465-466. Zbl0123.04302MR158857

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.