Size Lower Bounds for Quantum Automata

We compare the descriptional power of quantum finite automata with control language (qfcs) and deterministic finite automata (dfas). By suitably adapting Rabin’s technique, we show how to convert any given qfc to an equivalent dfa, incurring in an at most exponential size increase. This enables us to state a lower bound on the size of qfcs, which is logarithmic in the size of equivalent minimal dfas. In turn, this result yields analogous size lower bounds for several models of quantum finite automata in the literature.

[1]  Maksim Kravtsev,et al.  Probabilistic Reversible Automata and Quantum Automata , 2002, COCOON.

[2]  Andris Ambainis,et al.  Algebraic Results on Quantum Automata , 2005, Theory of Computing Systems.

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

[4]  Alberto Bertoni,et al.  Some formal tools for analyzing quantum automata , 2006, Theor. Comput. Sci..

[5]  Jeffrey D. Ullman,et al.  Introduction to automata theory, languages, and computation, 2nd edition , 2001, SIGA.

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

[7]  Carlo Mereghetti,et al.  Quantum finite automata with control language , 2006, RAIRO Theor. Informatics Appl..

[8]  Andris Ambainis,et al.  Superiority of exact quantum automata for promise problems , 2011, Inf. Process. Lett..

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

[10]  Ashwin Nayak,et al.  Optimal lower bounds for quantum automata and random access codes , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[11]  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).

[12]  Mika Hirvensalo,et al.  Quantum Automata with Open Time Evolution , 2010, Int. J. Nat. Comput. Res..

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

[14]  Azaria Paz,et al.  Probabilistic automata , 2003 .

[15]  Beatrice Palano,et al.  Behaviours of Unary Quantum Automata , 2010, Fundam. Informaticae.

[16]  A. C. Cem Say,et al.  Unbounded-error quantum computation with small space bounds , 2010, Inf. Comput..

[17]  Alberto Bertoni,et al.  Small size quantum automata recognizing some regular languages , 2005, Theor. Comput. Sci..

[18]  Beatrice Palano,et al.  Events and Languages on Unary Quantum Automata , 2009, NCMA.

[19]  K. Scharnhorst,et al.  Angles in Complex Vector Spaces , 1999 .

[20]  R. Hughes,et al.  The Structure and Interpretation of Quantum Mechanics , 1989 .

[21]  Shenggen Zheng,et al.  One-Way Finite Automata with Quantum and Classical States , 2011, Languages Alive.