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arXiv:2212.14442 · Lobachevskii Journal of Mathematics

Deterministic Construction of QFAs Based on the Quantum Fingerprinting Technique

Aliya Khadieva, Mansur Ziatdinov

Published Feb 1, 2023• 6 citations
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Abstract

It is known that for some languages quantum finite automata are more efficient than classical counterparts. Particularly, a QFA recognizing the language $$MOD_{p}$$ has an exponential advantage over the classical finite automata. However, the construction of such QFA is probabilistic. In the current work, we propose a deterministic construction of the QFA for the language $$MOD_{p}$$ . We construct a QFA for a promise problem $$Palindrome_{s}$$ and implement this QFA on the IBMQ simulator using qiskit library tools.

DOI
10.1134/s199508022302021x

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It is known that for some languages quantum finite automata are more efficient than classical counterparts. Particularly, a QFA recognizing the language MODpMOD_{p}MODp​ has an exponential advantage over the classical finite automata. However, the construction of such QFA is probabilistic. In the current work, we propose a deterministic construction of the QFA for the language MODpMOD_{p}MODp​ . We construct a QFA for a promise problem PalindromesPalindrome_{s}Palindromes​ and implement this QFA on the IBMQ simulator using qiskit library tools.
We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error QFAs are more powerful than PFAs. But, in contrary to the binary problems, the computational powers of Las-Vegas QFAs and bounded-error PFAs are equivalent to deterministic finite automata (DFAs). Lastly, we present a new family of unary promise problems with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentially more succinct than DFAs.
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 MODp={aj ∣ j≡0mod  p}\mathtt{MOD}_{\rm p} = \{a^{j}~ |~ j \equiv 0 \mod p\}MODp​={aj ∣ j≡0modp}, where ppp is a prime number. In this paper, we present a modified MCQFA algorithm for the language MODp\mathtt{MOD}_{\rm p}MODp​, 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.
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 MODp={aj∣j≡0mod  p}\mathtt{MOD}_p = \{ a^{j} \mid j \equiv 0 \mod p\}MODp​={aj∣j≡0modp}, where ppp is a prime number. In this paper, we present a modified MCQFA algorithm for the language MODp\mathtt{MOD}_pMODp​, 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.
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.