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arXiv:2407.02776

A Framework for Quantum Finite-State Languages with Density Mapping

SeungYeop Baik, Sicheol Sung, Yo-Sub Han

Published Jul 3, 2024• 0 citations

Abstract

A quantum finite-state automaton (QFA) is a theoretical model designed to simulate the evolution of a quantum system with finite memory in response to sequential input strings. We define the language of a QFA as the set of strings that lead the QFA to an accepting state when processed from its initial state. QFAs exemplify how quantum computing can achieve greater efficiency compared to classical computing. While being one of the simplest quantum models, QFAs are still notably challenging to construct from scratch due to the preliminary knowledge of quantum mechanics required for superimposing unitary constraints on the automata. Furthermore, even when QFAs are correctly assembled, the limitations of a current quantum computer may cause fluctuations in the simulation results depending on how an assembled QFA is translated into a quantum circuit. We present a framework that provides a simple and intuitive way to build QFAs and maximize the simulation accuracy. Our framework relies on two methods: First, it offers a predefined construction for foundational types of QFAs that recognize special languages MOD and EQU. They play a role of basic building blocks for more complex QFAs. In other words, one can obtain more complex QFAs from these foundational automata using standard language operations. Second, we improve the simulation accuracy by converting these QFAs into quantum circuits such that the resulting circuits perform well on noisy quantum computers. Our framework is available at https://github.com/sybaik1/qfa-toolkit.

DOI: 10.48550/arxiv.2407.02776

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The study of quantum finite automata (QFA's) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as

\mathtt{MOD}_{\rm p} = \{a^{j}~ |~ j \equiv 0 \mod p\}

, where

p

is a prime number. In this paper, we present a modified MCQFA algorithm for the language

\mathtt{MOD}_{\rm p}

, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using less basis gates compared to the implementation of the original algorithm given in the literature.

We develop a quantum algorithm to verify the correctness of systems modeled by nondeterministic finite-state automata-a problem that is computationally intractable on conventional computers. Specifically, our algorithm solves the language containment problem in three steps. First, we translate a nondeterministic finite-state automaton into a quantum finite state automaton (QFA) circuit. Second, the QFA is embedded into a larger circuit which generates a superposed set of all bounded strings that are marked to indicate whether they are accepted or not. Finally, we amplify the amplitudes of accepted strings so that they are measured more frequently. The last step may be done either by using Grover's algorithm, or by using FnR, a custom algorithm that we have developed for the cases where Grover's algorithm is not effective. Our work represents the first proposal to apply quantum computing to the problem of verifying conventional systems; our approach would facilitate software verification, program analysis, protocol design, and verification of circuits, among other applications.

The study of quantum finite automata (QFAs) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as

\mathtt{MOD}_p = \{ a^{j} \mid j \equiv 0 \mod p\}

, where

p

is a prime number. In this paper, we present a modified MCQFA algorithm for the language

\mathtt{MOD}_p

, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using fewer basis gates compared to the implementation of the original algorithm given in the literature.

We study 1-way quantum finite automata (QFAs). First, we compare them with their classical counterparts. We show that, if an automaton is required to give the correct answer with a large probability (greater than 7/9), then any 1-way QFAs can be simulated by a 1-way reversible automaton. However, quantum automata giving the correct answer with smaller probabilities are more powerful than reversible automata. Second, we show that 1-way QFAs can be very space-efficient. We construct a 1-way QFA that is exponentially smaller than any equivalent classical (even randomized) finite automaton. We think that this construction may be useful for design of other space-efficient quantum algorithms. Third, we consider several generalizations of 1-way QFAs. Here, our goal is to find a model which is more powerful than 1-way QFAs keeping the quantum part as simple as possible.

The 1-way quantum finite automata and 2-way quantum finite automata are the first ever developed quantum automatons. But they suffer from many disadvantages which make them less powerful and impracticable. To remove these drawbacks there are many ways to modify and aggrandize the quantum finite automatons. Including 1-counter with 1-way QFA significantly improve its power. Likely quantum finite automata with quantum and classical states (QCFA) and 2-way quantum finite 1-counter automata (2-QF1-CA) are the quintessence of these modifications. In our paper we first discuss about 1-QFA and 2-QFA and their drawbacks. In the section IV we introduce an algorithm by implementing 1-way quantum finite k-counter automata (1-QFk-CA). Then we discuss another algorithm by implementing quantum finite 1-counter automata (QF1-CA) with multiple scanning of the input string. These algorithms improve the power of QFAs.