Emulators of Quantum Computers on Qubits and on Qudits
Abstract
Quantum computing is still a developing, but an extremely promising area. The article lays out the main ideas behind quantum computing in simple terms. The topic of quantum computers based on qudits - multidimensional analogues of qubits, which have recently received much attention due to their efficiency, is also covered.
The fundamentals of quantum mechanics, which are necessary for understanding the principles of operation of a quantum computer, such concepts as qubits and qudits, linear operators, the measurement process, etc are introduced. As an example of quantum computing, the principle of operation of the Deutsch-Jozsa algorithm, one of the first quantum algorithms to demonstrate their advantages, and its generalization to qudits, are analyzed in detail.
The process of writing the simplest quantum computer emulator in the Python programming language is described step by step. The emulator operates with an arbitrary number of qubits and allows you to apply arbitrary operators to them and carry out multiple measurements of the final state of the qubit. A generalization of this emulator for working with qudits is given after that.
To demonstrate the emulator we have written, we present programs that implement the Deutsch-Jozsa algorithm and its generalizations on it, and test them.
References
2. Baskakov P.E., Khabovets Y.Yu., Pilipenko I.A., Kravchenko V.O., Cherkesova L.V. Tools for Performing and Emulating Quantum Computing. Vestnik Novosibirskogo gosudarstvennogo universiteta. Seriâ: informacionnye tehnologii v obrazovanii = Vestnik NSU. Series: Information Technologies. 2020; 18(2):43-53. (In Russ., abstract in Eng.) https://doi.org/10.25205/1818-7900-2020-18-2-43-53
3. Kiktenko E.O., Nikolaeva A.S., Fedorov A.K. Quantum computing using multilevel quantum systems. Nanoindustry. 2020; 13(S4):649-651. (In Russ., abstract in Eng.) doi: https://doi.org/10.22184/1993-8578.2020.13.4s.649.651
4. Smirnova T.S., Shvetskiy M.V. A visual emulator of the Bloch vector and sphere as a means of teaching quantum computing. The Scientific Opinion. 2021; (9):76-82. (In Russ., abstract in Eng.) doi: https://doi.org/10.25807/22224378_2021_9_76
5. Grigoryeva G.M., Khodchenkov V.Yu. On the possibility of building a quantum computer emulator using XMM registers. Sistemy komp’yuternoj matematiki i ih prilozheniya = Computer Mathematics Systems and Their Applications. 2021; (22):113-116. Available at: https://www.elibrary.ru/item.asp?id=46649884 (accessed 16.03.2022). (In Russ., abstract in Eng.)
6. Arute F., Arya K., Babbush R., et al. Quantum supremacy using a programmable superconducting processor. Nature. 2019; 574:505-510. (In Eng.) doi: https://doi.org/10.1038/s41586-019-1666-5
7. Guzik V.P., Gushanskiy S.M. Development of emulator for quantum computers. Izvestiya SFedU. Engineering Sciences. 2010; (2):73-79. Available at: https://www.elibrary.ru/item.asp?id=13617268 (accessed 16.03.2022). (In Russ., abstract in Eng.)
8. Solovyev V.M. Quantum Computers and Quantum Algorithms. Part 1. Quantum Computers. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics. 2015; 15(4):462-477. (In Russ., abstract in Eng.) doi: https://doi.org/10.18500/1816-9791-2015-15-4-462-477
9. Solovyev V.M. Quantum Computers and Quantum Algorithms. Part 2. Quantum Algorithms. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics. 2016; 16(1):104-112. (In Russ., abstract in Eng.) doi: https://doi.org/10.18500/1816-9791-2016-16-1-104-112
10. Ladd T., Jelezko F., Laflamme R., et al. Quantum computers. Nature. 2010; 464:45-53. (In Eng.) doi: https://doi.org/10.1038/nature08812
11. Kiktenko E.O., Fedorov A.K., Man'ko O.V., Man'ko V.I. Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities. Physical Review A. 2015; 91(4):042312. (In Eng.) doi: https://doi.org/10.1103/PhysRevA.91.042312
12. Imany P., Jaramillo-Villegas J.A., Alshaykh M.S., et al. High-dimensional optical quantum logic in large operational spaces. npj Quantum Information. 2019; 5:59. (In Eng.) doi: https://doi.org/10.1038/s41534-019-0173-8
13. Wang Y., Hu Z., Sanders B.C., Kais S. Qudits and High-Dimensional Quantum Computing. Frontiers in Physics. 2020; 8:589504. (In Eng.) doi: https://doi.org/10.3389/fphy.2020.589504
14. Kiktenko E.O., Nikolaeva A.S., Xu P., Shlyapnikov G.V., Fedorov A.K. Scalable quantum computing with qudits on a graph. Physical Review A. 2020; 101(2):022304. (In Eng.) doi: https://doi.org/10.1103/PhysRevA.101.022304
15. Moreno-Pineda E., Godfrin C., Balestro F., Wernsdorfer W., Ruben M. Molecular spin qudits for quantum algorithms. Chemical Society Reviews. 2018; 47(2), 501-513. (In Eng.) doi: https://doi.org/10.1039/C5CS00933B
16. Ringbauer M., Meth M., Postler L., Stricker R., Blatt R., Schindler P., Monz T. A universal qudit quantum processor with trapped ions. Nature Physics. 2022; 18:1053-1057. (In Eng.) doi: https://doi.org/10.1038/s41567-022-01658-0
17. Tacchino F., Chiesa A., Sessoli R., Tavernelli I., Carretta S. A proposal for using molecular spin qudits as quantum simulators of light–matter interactions. Journal of Materials Chemistry C. 2021; 9(32):10266-10275. (In Eng.) doi: https://doi.org/10.1039/D1TC00851J
18. Lu H.H., Hu Z., Alshaykh M.S., Moore A.J., Wang Y., Imany P., Weiner A.M., Kais S. Quantum Phase Estimation with Time-Frequency Qudits in a Single Photon. Advanced Quantum Technologies. 2020; 3(2):1900074. (In Eng.) doi: https://doi.org/10.1002/qute.201900074
19. Fischer L.E., Chiesa A., Tacchino F., Egger D.J., Carretta S., Tavernelli I. Towards universal gate synthesis and error correction in transmon qudits. arXiv:2212.04496. 2022. (In Eng.) doi: https://doi.org/10.48550/arXiv.2212.04496
20. Chi Y., Huang J., Zhang Z., et al. A programmable qudit-based quantum processor. Nature Communications. 2022; 13:1166. (In Eng.) doi: https://doi.org/10.1038/s41467-022-28767-x
21. Brennen G.K., O'Leary D.P., Bullock S.S. Criteria for exact qudit universality. Physical Review A. 2005; 71(5):052318. (In Eng.) doi: https://doi.org/10.1103/PhysRevA.71.052318
22. Biamonte J., Wittek P., Pancotti N., et al. Quantum machine learning. Nature. 2017; 549:195-202. (In Eng.) doi: https://doi.org/10.1038/nature23474
23. Aryte F., Arya K., Babbush R., et al. Quantum supremacy using a programmable supercon-ducting processor. Nature. 2019; 574:505-510. (In Eng.) doi: https://doi.org/10.1038/s41586-019-1666-5
24. Deutsch D., Jozsa R. Rapid Solution of Problems by Quantum Computation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 1992; 439(1907):553-558. (In Eng.) doi: https://doi.org/10.1098/rspa.1992.0167
25. Fan Y. A Generalization of the Deutsch-Jozsa Algorithm to Multi-Valued Quantum Logic. 37th International Symposium on Multiple-Valued Logic (ISMVL'07). IEEE Computer Society, Oslo, Norway; 2007. p. 1-5. (In Eng.) doi: https://doi.org/10.1109/ISMVL.2007.3
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
