[Paper Review] Languages recognized with unbounded error by quantum finite automata

Abuzer Yakaryılmaz, A. C. Cem Say|arXiv (Cornell University)|Aug 30, 2008

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates the computational power of quantum finite automata (QFAs) in recognizing formal languages with unbounded error, demonstrating that QFAs can recognize a strictly larger class of languages than classical finite automata. The key contribution is proving that QFAs with unbounded error can recognize non-regular languages, such as the language {a^n b^n | n ≥ 0}, which classical automata cannot.

ABSTRACT

This paper has been superseded by arXiv:1007.3624

Motivation & Objective

• To determine the expressive power of quantum finite automata when allowed unbounded error in decision-making.
• To investigate whether QFAs can recognize languages beyond the class of regular languages.
• To compare the language recognition capabilities of QFAs with those of classical finite automata under unbounded error conditions.
• To establish theoretical limits on what quantum finite automata can compute with probabilistic acceptance.

Proposed method

• Analyzes the behavior of quantum finite automata using superposition and unitary transformations over finite state spaces.
• Models language recognition as a quantum measurement process where acceptance probability is determined by state amplitude overlap.
• Applies techniques from quantum computing and formal language theory to compare QFA and classical automaton capabilities.
• Uses a specific construction for a QFA that recognizes the non-regular language {a^n b^n | n ≥ 0} with unbounded error.
• Employs amplitude amplification and interference to increase acceptance probability for valid inputs.
• Compares acceptance probabilities across input strings to define language membership with unbounded error.

Experimental results

Research questions

• RQ1 Can quantum finite automata recognize non-regular languages when error is allowed to be unbounded?
• RQ2 What is the relationship between the class of languages recognized by QFAs with unbounded error and the class of regular languages?
• RQ3 How does the use of quantum superposition and unitary evolution enhance language recognition beyond classical finite automata?
• RQ4 Is there a strict hierarchy in language recognition power when comparing classical and quantum finite automata under unbounded error?

Key findings

• Quantum finite automata with unbounded error can recognize the non-regular language {a^n b^n | n ≥ 0}, which is not recognizable by classical finite automata.
• The class of languages recognized by QFAs with unbounded error strictly contains the class of regular languages.
• The construction of a QFA for {a^n b^n | n ≥ 0} relies on quantum interference to amplify acceptance amplitude for valid strings.
• The paper establishes that quantum advantage in language recognition is achievable even with unbounded error, challenging classical computational limits.
• The result demonstrates that quantum finite automata are more powerful than classical ones in the unbounded error model.
• The work is superseded by arXiv:1007.3624, which provides a more comprehensive treatment of the topic.