Abstract
We present a framework for the implementation of quantum finite automata algorithms designed for the language \( \mathtt {MOD_p} = \{ a^{i \cdot p} \mid i \ge 0 \} \) on gate-based quantum computers. First, we compile the known theoretical results from the literature to reduce the number of CNOT gates. Second, we demonstrate techniques for modifying the algorithms based on the basis gates of available quantum hardware in order to reduce circuit depth. Lastly, we explore how the number of CNOT gates may be reduced further if the topology of the qubits is known.
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Notes
- 1.
As the control operators are fired when a qubit is in state \(\left| 1\right\rangle \), we use several NOT (X) gates between controlled rotations. For example, we apply NOT gates on the first \(\log d\) qubits before and after the controlled \(R_{k_1}\) operator, and, in this way, we guarantee that \(R_{k_1}\) is fired only if the first \(\log d\) qubits are in \( \left| 0 \cdots 0\right\rangle \). If we follow the order of indices on the circuit, then there will be several NOT gates. But, if we follow an order based on Gray code, then it will be enough to use only a single NOT gate between the controlled rotations.
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Acknowledgments
We sincerely thank our colleagues Kamil Khadiev, Mansur Ziatdinov, Aleksander Vasiliev, and Aida Gainutdinova for useful discussions. Part of this work was done by Khadevia during QCourse570-1 “Projects in Quantum” in Spring 2022 conducted by QWorld & University of Latvia and supported by Unitary Fund. The research in Sect. 3 has been supported by the Kazan Federal University Strategic Academic Leadership Program (“PRIORITY-2030”). The research in Sects. 4 and 5 is supported by Russian Science Foundation Grant 24-21-00406, https://rscf.ru/en/project/24-21-00406/.
Salehi was partially supported by Polish National Science Center under the grant agreement 2019/33/B/ST6/02011.
Yakaryılmaz was partially supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1 /22/I/CFLA/001, the ERDF project Nr. 1.1.1.5/19/A/005 “Quantum computers with constant memory”, and the ERDF project number 1.1.1.5/18/A/020 “Quantum algorithms: from complexity theory to experiment”.
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Khadieva, A., Salehi, Ö., Yakaryılmaz, A. (2024). A Representative Framework for Implementing Quantum Finite Automata on Real Devices. In: Cho, DJ., Kim, J. (eds) Unconventional Computation and Natural Computation. UCNC 2024. Lecture Notes in Computer Science, vol 14776. Springer, Cham. https://doi.org/10.1007/978-3-031-63742-1_12
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