Automata and Quantum Computing (1507.01988v2)

Abstract: Quantum computing is a new model of computation, based on quantum physics. Quantum computers can be exponentially faster than conventional computers for problems such as factoring. Besides full-scale quantum computers, more restricted models such as quantum versions of finite automata have been studied. In this paper, we survey various models of quantum finite automata and their properties. We also provide some open questions and new directions for researchers. Keywords: quantum finite automata, probabilistic finite automata, nondeterminism, bounded error, unbounded error, state complexity, decidability and undecidability, computational complexity

• The paper provides a comprehensive survey of various quantum finite automata (QFA) models, including 1QFA, 2QFA, and AQFA, analyzing their computational power and language recognition capabilities.
• It highlights QFAs can be exponentially more succinct than classical finite automata in state complexity for recognizing certain languages, demonstrating advantages in scenarios like unary languages.
• The survey discusses practical implications for compact computation, raises theoretical questions about computational classes, and identifies open problems for future research in quantum automata theory.

Overview of Automata and Quantum Computing

The paper by Andris Ambainis and Abuzer Yakaryılmaz titled "Automata and Quantum Computing" provides a comprehensive survey of the intersection between automata theory and quantum computing. This paper explores various models of quantum finite automata (QFAs) and seeks to understand their capabilities, limitations, and implications.

Models and Definitions

The paper introduces several key models of quantum finite automata, which are pivotal to understanding quantum computation in automata:

1. One-Way Quantum Finite Automata (1QFA): This model forms the foundation of QFA research, with variations such as Moore-Crutchfield, Kondacs-Watrous, Latvian, and Nayak versions that incorporate differing combinations of unitary transformations and quantum measurements.
2. Two-Way Quantum Finite Automata (2QFA): A more complex model that extends the one-way versions to allow head movement in both directions in the input tape, enhancing computational power but also increasing complexity in analysis.
3. Alternating Quantum Finite Automata (AQFA): Extending the nondeterministic model, AQFAs are highlighted for their capability to recognize more complex languages, such as PSPACE-complete problems.

The paper details the computational abilities of these models, emphasizing that QFAs can be exponentially more succinct than deterministic and probabilistic finite automata in some cases, particularly over unary language alphabets.

Language Recognition and Succinctness

Ambainis and Yakaryılmaz delve into the various language classes recognizable by QFAs under different constraints:

• Bounded Error: The paper illustrates how certain language classes are recognizable by QFAs with bounded error, showing them to be more powerful than classical models in specific scenarios. It introduces elegant results, such as recognizing the language EQ with polynomial expected time, a feat generally beyond the scope of 2PFA.
• Unbounded Error and Nondeterminism: In this context, QFAs have demonstrated equivalence with PFAs in recognizing stochastic languages. The characterization of 1NQFA as accepting the class $\mathsf{S}^{\neq}$ presents significant implications for the computational theory.
• State Complexity and Succinctness: A highlight of the paper is showcasing scenarios where QFAs hold a substantial advantage in state complexity, being able to recognize languages that classical automata can only describe with vastly more states.

Practical and Theoretical Implications

The practical implications of these findings point towards utilizing QFAs in areas requiring compact representations of automata—such as memory-limited computational environments. From a theoretical perspective, the survey raises questions about the completeness of certain classes and potential future breakthroughs in quantum automata capabilities.

Future Directions

The paper identifies several open questions and future research directions, including:

• Examining the full potential of two-way QFAs.
• Further analyzing the computational limits of alternating quantum models.
• Investigating more deeply the algebraic properties of QFAs and their computational advantages.

Conclusion

Through a formal survey, Ambainis and Yakaryılmaz offer significant insight into quantum finite automata, providing the research community with a large number of mathematical models, concrete examples, and practical challenges. The intersection of automata theory and quantum computing remains ripe for exploration, promising advancements in both theoretical underpinnings and practical implementations in quantum technologies.