Quantum finite automata: A modern introduction (1406.4048v1)

Abstract: We present five examples where quantum finite automata (QFAs) outperform their classical counterparts. This may be useful as a relatively simple technique to introduce quantum computation concepts to computer scientists. We also describe a modern QFA model involving superoperators that is able to simulate all known QFA and classical finite automaton variants.

• The paper provides a comprehensive review of quantum finite automata (QFAs), proposing a modern model using superoperators and highlighting their advantages over classical automata with key examples.
• QFAs exhibit superior performance over classical finite automata (FAs) in examples like recognizing unary languages with uncountably infinite cutpoints or recognizing nonregular languages with a cutpoint of 0.
• The modern QFA model can simulate all known variants of QFAs and classical automata, offering insights into succinct solutions for promise problems and efficient bounded-error recognition of certain languages.

Quantum Finite Automata: A Modern Introduction

This paper presents a comprehensive review of quantum finite automata (QFAs) by demonstrating their capabilities compared to classical finite automata, highlighting key examples where QFAs show superior performance. It examines fundamental concepts and various models of QFAs, providing insights into their utility as a pedagogical tool for introducing quantum computation concepts to computer scientists.

The authors begin by acknowledging the limitations of early QFA models, which did not utilize the full power of quantum mechanics, leading to confusion when quantum machines could not simulate classical counterparts. They propose a modern model of QFAs involving superoperators, which facilitate the simulation of all known QFA and classical automaton variants.

Key Examples Demonstrating QFA Superiority

Five primary examples illustrate how QFAs surpass classical finite automata:

1. Recognition of Unary Languages with Cutpoints: The authors explain how a simple two-state QFA can define uncountably infinitely many tally languages with different cutpoints, unlike classical probabilistic finite automata (PFAs), which can define only regular languages with cutpoints.
2. Nondeterministic Quantum Recognition of Nonregular Languages: The paper highlights how QFAs, unlike nondeterministic classical automata, can recognize nonregular languages with a cutpoint of 0, providing a concrete example of the language $\mathtt{NEQ}$, which consists of strings where the number of two specific symbols are not equal.
3. Succinct Solution of Promise Problems: The authors demonstrate the efficient use of QFAs in solving promise problems exactly, showcasing significant reductions in the number of states required compared to deterministic finite automata (DFAs).
4. Succinct Bounded-Error Language Recognition: An example of QFAs recognizing the language $\mathtt{MOD^p}$ with bounded error using fewer states than classical PFAs is presented, emphasizing the exponential state gap between QFAs and PFAs.
5. Bounded-Error Recognition of Nonregular Languages in Polynomial Time: The authors construct a two-way QFA recognizing the nonregular language $\mathtt{EQ}$ efficiently in polynomial time, a feat not possible with two-way PFAs.

Theoretical and Practical Implications

The implications of these findings are profound in both the theoretical understanding and practical application of QFAs in quantum computing. The authors stress that QFAs could serve as a gateway to understanding larger quantum systems, such as quantum Turing machines and quantum circuits, due to their relative simplicity yet powerful capabilities. The examples also hint at potential applications in fields where succinct state representation and probabilistic computations are critical.

Future Directions

Looking forward, the paper proposes further exploration into the classes of languages QFAs can efficiently recognize and the potential for new QFA models that leverage advanced quantum mechanics properties. It invites researchers to investigate the limits and extensions of QFAs, especially concerning succinctness and efficiency compared to other computational paradigms.

In summary, this paper establishes quantum finite automata as a compelling and versatile subject within quantum computing research, offering valuable insights into their advantages and future potential. By systematically examining QFAs, the authors provide a foundation for further development and integration of quantum automata into practical and theoretical computing frameworks.