Postselection Finite Quantum Automata

Postselection for quantum computing devices was introduced by S. Aaronson[2] as an excitingly efficient tool to solve long standing problems of computational complexity related to classical computing devices only. This was a surprising usage of notions of quantum computation. We introduce Aaronson's type postselection in quantum finite automata. There are several nonequivalent definitions of quantumfinite automata. Nearly all of them recognize only regular languages but not all regular languages. We prove that PALINDROMES can be recognized by MM-quantum finite automata with postselection. At first we prove by a direct construction that the complement of this language can be recognized this way. This result distinguishes quantum automata from probabilistic automata because probabilistic finite automata with non-isolated cut-point 0 can recognize only regular languages but PALINDROMES is not a regular language.

[1]  Andris Ambainis,et al.  Dense quantum coding and quantum finite automata , 2002, JACM.

[2]  Rusins Freivalds,et al.  Why Sometimes Probabilistic Algorithms Can Be More Effective , 1996, MFCS.

[3]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[4]  Rusins Freivalds.  Projections of Languages Recognizable by Probabilistic and Alternating Finite Multitape Automata , 1981, Inf. Process. Lett..

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

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

[7]  Scott Aaronson,et al.  Lower bounds for local search by quantum arguments , 2003, STOC '04.

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

[9]  Antoni Kreczmar,et al.  Mathematical Foundations of Computer Science 1989 , 1989, Lecture Notes in Computer Science.

[10]  Andris Ambainis,et al.  Probabilities to Accept Languages by Quantum Finite Automata , 1999, COCOON.

[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]  Dines Bjørner,et al.  Baltic Computer Science , 1991, Lecture Notes in Computer Science.

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

[14]  Farid M. Ablayev.  On Comparing Probabilistic and Deterministic Automata Complexity of Languages , 1989, MFCS.

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

[16]  Rusins Freivalds.  Non-constructive Methods for Finite Probabilistic Automata , 2007, Developments in Language Theory.

[17]  Andris Ambainis,et al.  Exact Results for Accepting Probabilities of Quantum Automata , 2001, MFCS.

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

[19]  Rusins Freivalds.  Complexity of Probabilistic Versus Deterministic Automata , 1991, Baltic Computer Science.

[20]  Ronald de Wolf,et al.  Exponential lower bound for 2-query locally decodable codes via a quantum argument , 2002, STOC '03.

[21]  Dorit Aharonov,et al.  Lattice problems in NP ∩ coNP , 2005, JACM.

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

[23]  Ioan I. Macarie.  Space-Efficient Deterministic Simulation of Probabilistic Automata , 1998, SIAM J. Comput..

[24]  Dorit Aharonov,et al.  Lattice Problems in NP cap coNP , 2004, FOCS.

[25]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.