A Framework for Quantum Finite-State Languages with Density Mapping

TL;DR

This work addresses the challenge of designing and simulating quantum finite-state languages (QFLs) with limited qubits by introducing a structured framework that builds complex QFLs from simple building blocks (MOD and EQU) and improves simulation accuracy through a density-mapping technique that guides transpilation to quantum circuits. It formalizes the MO-QFA and MM-QFA models, their language classes, and closure properties, and details how Boolean operations compose QFLs. The key contributions are the formal two-building-block approach, the D-mapping method to reduce bit-flip errors during circuit translation, and experimental validation showing improved simulator accuracy and real-device limitations. The framework advances practical QFA design for near-term quantum hardware and lays groundwork for broader language construction and improved error mitigation in quantum automata.

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.

Figures (5)

• Figure 1: The three main features of our framework are as follows: basic blocks, language operations and density mapping. Each feature is presented in (a) a formal description, (b) a circuit and (c) a graphical representation.
• Figure 2: The relation between (a) a unitary transition matrix of a semi-QFA, (b) a change of superposition on the Bloch sphere according to a transition and (c) a graphical representation of the matrix's transitions. Transitions of QFAs, for example $\ket{q_0} \mapsto U_a\ket{q_0}$, are represented as a unitary transformation in Hilbert space.
• Figure 3: The expected error reduction of the proposed $\mathcal{D}$-mapping. The accepting state $q_1$ is mapped to the qubit basis $\ket{10000}$. If a bit-flip error occurs on the fourth qubit, then the measured qubit basis is $\ket{10010}$ that describes the state $q_n$. It does not effect whether or not the string is accepted because both $q_1$ and $q_k$ are accepting.
• Figure 4: The experimental results for the language ${\mathsf{MOD}}_5$. Blue circles and orange squares indicate the accepting probability of strings, which are simulated on a real-world quantum computer and a simulator, respectively. The dashed line represents the 0.5 mark for random guessing. We set the mirroring factors $\eta_g$ and $\eta_r$ as 0.5%. In the case of an ideal computer with no errors, the accepting probabilities highlighted by gray boxes are 1.
• Figure 5: The difference in accepting probability from using the naive mapping to using $\mathcal{D}$-mapping on ${\mathsf{MOD}}_5$. We set the mirroring factors $\eta_g$ and $\eta_r$ as 0.5%. The blue dots above the red line represent the increase in accepting probability of $\mathcal{D}$-mapping compared to the naive mapping. Conversely, the blue dots below the red line represent the decrease in accepting probability. The strings highlighted with gray boxes should exhibit an increased accepting probability towards the ideal case, while other strings should show a decrease.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 2
• proof