kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-2-95-105kaz29-1257Research ArticleКОМПЬЮТЕРНЫЕ НАУКИCOMPUTER SCIENCEИНВАРИАНТНЫЕ МНОГОЧЛЕНЫ С ПРИМЕНЕНИЕМ К КВАНТОВЫМ ВЫЧИСЛЕНИЯМINVARIANT POLYNOMIALS WITH APPLICATIONS TO QUANTUM COMPUTINGhttps://orcid.org/0000-0002-7669-0656АмановА.AmanovA.

PhD студент

050000, г. Алматы

PhD student

050000, Almaty

alimzhan.amanov@gmail.comКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstan20240107202421295105Copyright © Аманов А., 20242024Аманов А.Amanov A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/1257

В теории квантовой информации понимание сложности запутанных состояний в контексте SLOCC (стохастические локальные операции и классическая коммуникация) с d кубитами (или кудитами) является важным для продвижения наших знаний о квантовых системах. Эта сложность часто анализируется путем классификации состояний через локальные группы симметрии. Полученные классы можно различить с помощью инвариантных многочленов, которые служат мерой запутанности. В данной статье представлен новый метод получения инвариантных многочленов наименьших степеней, что значительно повышает эффективность характеристики классов SLOCC запутанных квантовых состояний. Наш метод не только упрощает процесс идентификации этих классов, но и предоставляет надежный инструмент для анализа свойств запутанности сложных квантовых систем. В качестве практического применения мы демонстрируем вывод инвариантов минимальной степени в специальных случаях, иллюстрируя эффективность нашего подхода в реальных сценариях. Это достижение имеет потенциал для упрощения различных процессов в теории квантовой информации, делая понимание, классификацию и использование запутанных состояний более легкими и эффективными.

In quantum information theory, understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving d qubits (or qudits) is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. The resulting classes can be distinguished using invariant polynomials, which serve as a measure of entanglement. This paper introduces a novel method for obtaining invariant polynomials of the smallest degrees, which significantly enhances the efficiency of characterizing SLOCC classes of entangled quantum states. Our method not only simplifies the process of identifying these classes but also provides a robust tool for analyzing the entanglement properties of complex quantum systems. As a practical application, we demonstrate the derivation of minimal degree invariants in specific cases, illustrating the effectiveness of our approach in real-world scenarios. This advancement has the potential to streamline various processes in quantum information theory, making it easier to understand, classify, and utilize entangled states effectively.

инвариантные полиномыквантовая запутанностьSLOCCinvariant polynomialsquantum entanglementSLOCCThis research was supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14869221).This research was supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14869221).ReferencesDür W., Vidal G. & Cirac J.I. (2000). Three qubits can be entangled in two inequivalent ways. Physical Review A, 62(6), 062314.Dür W., Vidal G. & Cirac J.I. (2000). Three qubits can be entangled in two inequivalent ways. Physical Review A, 62(6), 062314.J.-G. Luque and Jean-Yves Thibon. (2003) Polynomial invariants of four qubits. Physical Review A, vol. 67, no.J.-G. Luque and Jean-Yves Thibon. (2003) Polynomial invariants of four qubits. Physical Review A, vol. 67, no.https://doi.org/10.1103/physreva.67.042303. 3 J.-G. Luque and Jean-Yves Thibon. (2005) Algebraic invariants of five qubits. Journal of physics, vol. 39, no. 2, pp. 371–377. https://doi.org/10.1088/0305-4470/39/2/007.https://doi.org/10.1103/physreva.67.042303. 3 J.-G. Luque and Jean-Yves Thibon. (2005) Algebraic invariants of five qubits. Journal of physics, vol. 39, no. 2, pp. 371–377. https://doi.org/10.1088/0305-4470/39/2/007.Horodecki R., Horodecki P., Horodecki M. and Horodecki K. (2009) Quantum entanglement," Reviews of Modern Physics, vol. 81, no. 2, pp. 865–942. https://doi.org/10.1103/revmodphys.81.865.Horodecki R., Horodecki P., Horodecki M. and Horodecki K. (2009) Quantum entanglement," Reviews of Modern Physics, vol. 81, no. 2, pp. 865–942. https://doi.org/10.1103/revmodphys.81.865.Nielsen M.A. and Chuang I.L. (2019) Quantum computation and quantum information. Cambridge Cambridge University Press.Nielsen M.A. and Chuang I.L. (2019) Quantum computation and quantum information. Cambridge Cambridge University Press.Hillery M., Buâek V. & Berthiaume A. (1999) Quantum secret sharing. Physical Review A, 59(3), 1829.Hillery M., Buâek V. & Berthiaume A. (1999) Quantum secret sharing. Physical Review A, 59(3), 1829.Miyake A. (2003) Classification of multipartite entangled states by multidimensional determinants, Physical Review A, vol. 67, no. 1. https://doi.org/10.1103/physreva.67.012108.Miyake A. (2003) Classification of multipartite entangled states by multidimensional determinants, Physical Review A, vol. 67, no. 1. https://doi.org/10.1103/physreva.67.012108.Bürgisser P. and C. Ikenmeyer. (2017) Fundamental invariants of orbit closures. Journal of Algebra, vol. 477, pp. 390–434. https://doi.org/10.1016/j.jalgebra.2016.12.035.Bürgisser P. and C. Ikenmeyer. (2017) Fundamental invariants of orbit closures. Journal of Algebra, vol. 477, pp. 390–434. https://doi.org/10.1016/j.jalgebra.2016.12.035.Bürgisser P., Garg A., OliveiraR., Walter M. and Wigderson A. (2018) Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 94, pp. 24:1–24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ITCS.2018.24.Bürgisser P., Garg A., OliveiraR., Walter M. and Wigderson A. (2018) Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 94, pp. 24:1–24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ITCS.2018.24.Amanov A. and Yeliussizov D. (2022) Fundamental Invariants of Tensors, Latin Hypercubes, and Rectangular Kronecker Coefficients, International Mathematics Research Notices, vol. 2023, no. 20, pp. 17552–17599. https://doi.org/10.1093/imrn/rnac311.Amanov A. and Yeliussizov D. (2022) Fundamental Invariants of Tensors, Latin Hypercubes, and Rectangular Kronecker Coefficients, International Mathematics Research Notices, vol. 2023, no. 20, pp. 17552–17599. https://doi.org/10.1093/imrn/rnac311.Bürgisser P., Garg A., Oliveira R., Walter M. and Wigderson A. (2018) Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 94, pp. 24:1–24:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ITCS.2018.24Bürgisser P., Garg A., Oliveira R., Walter M. and Wigderson A. (2018) Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 94, pp. 24:1–24:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ITCS.2018.24Blinder S.M. (2022) Three-Qubit W-States on a Quantum Computer, Wolfram Demonstrations Project. [Online]. Available: http://demonstrations.wolfram.com/ThreeQubitWStatesOnAQuantumC omputer/ (accessed: Jun. 14, 2024).Blinder S.M. (2022) Three-Qubit W-States on a Quantum Computer, Wolfram Demonstrations Project. [Online]. Available: http://demonstrations.wolfram.com/ThreeQubitWStatesOnAQuantumC omputer/ (accessed: Jun. 14, 2024).Cervera-Lierta A., Gasull A., Latorre J.I. and Sierra G. (2018) Multipartite entanglement in spin chains and the hyperdeterminant. Journal of physics. A, Mathematical and theoretical (Print), vol. 51, no. 50, pp. 505301–505301. https://doi.org/10.1088/1751-8121/aaee1f.Cervera-Lierta A., Gasull A., Latorre J.I. and Sierra G. (2018) Multipartite entanglement in spin chains and the hyperdeterminant. Journal of physics. A, Mathematical and theoretical (Print), vol. 51, no. 50, pp. 505301–505301. https://doi.org/10.1088/1751-8121/aaee1f.Viehmann O., Eltschka C. and Siewert J. (2011) Polynomial invariants for discrimination and classification of four-qubit entanglement. Physical Review A, vol. 83, no. 5. https://doi.org/10.1103/physreva.83.052330.Viehmann O., Eltschka C. and Siewert J. (2011) Polynomial invariants for discrimination and classification of four-qubit entanglement. Physical Review A, vol. 83, no. 5. https://doi.org/10.1103/physreva.83.052330.Cayley A. (1844) On the theory of determinants. Pitt Press. 16 Cayley A. (1845) On the theory of linear transformations. E. Johnson.Cayley A. (1844) On the theory of determinants. Pitt Press. 16 Cayley A. (1845) On the theory of linear transformations. E. Johnson.Gelfand I.M., Kapranov M.M. and Zelevinsky A.V. (1992) Hyperdeterminants, Advances in Mathematics, vol. 96, no. 2, pp. 226–263. https://doi.org/10.1016/0001-8708(92)90056-q.Gelfand I.M., Kapranov M.M. and Zelevinsky A.V. (1992) Hyperdeterminants, Advances in Mathematics, vol. 96, no. 2, pp. 226–263. https://doi.org/10.1016/0001-8708(92)90056-q.SageMath Mathematical Software System – Sage, SageMath Mathematical Software System. http://www.sagemath.org (accessed Apr. 04, 2024).SageMath Mathematical Software System – Sage, SageMath Mathematical Software System. http://www.sagemath.org (accessed Apr. 04, 2024).Maria C. (2021) Parameterized Complexity of Quantum Invariants, Proceedings of the 37th International Symposium on Computational Geometry (SoCG 2021). https://doi.org/10.4230/LIPIcs. SoCG.2021.53.Maria C. (2021) Parameterized Complexity of Quantum Invariants, Proceedings of the 37th International Symposium on Computational Geometry (SoCG 2021). https://doi.org/10.4230/LIPIcs. SoCG.2021.53.Haddadin W. (2021) Invariant polynomials and machine learning. arXiv preprint arXiv:2104.12733.Haddadin W. (2021) Invariant polynomials and machine learning. arXiv preprint arXiv:2104.12733.Liu J. (2019) Block distance invariant method for monoterm canonicalization of Riemann tensor polynomials, ACM Communications in Computer Algebra, vol. 53, pp. 134–137. https://doi.org/10.1145/3377006.3377019.Liu J. (2019) Block distance invariant method for monoterm canonicalization of Riemann tensor polynomials, ACM Communications in Computer Algebra, vol. 53, pp. 134–137. https://doi.org/10.1145/3377006.3377019.Avohou R.C., Geloun J.B. & Dub N. (2020) On the counting of O(N) tensor invariants. Advances in Theoretical and Mathematical Physics, vol. 24, no. 4. https://doi.org/10.4310/ATMP.2020.v24.n4.a1.Avohou R.C., Geloun J.B. & Dub N. (2020) On the counting of O(N) tensor invariants. Advances in Theoretical and Mathematical Physics, vol. 24, no. 4. https://doi.org/10.4310/ATMP.2020.v24.n4.a1.Geloun J.B. (2020) On the counting tensor model observables as U(N) and O(N) classical invariants, Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" - PoS(CORFU2019). https://doi.org/10.22323/1.376.0175.Geloun J.B. (2020) On the counting tensor model observables as U(N) and O(N) classical invariants, Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" - PoS(CORFU2019). https://doi.org/10.22323/1.376.0175.Grochow J.A. & Qiao Y. (2021) On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness, Leibniz International Proceedings in Informatics (LIPIcs), vol. 2021, pp. 31:1–31:19. https://doi.org/10.4230/LIPIcs.ITCS.2021.31.Grochow J.A. & Qiao Y. (2021) On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness, Leibniz International Proceedings in Informatics (LIPIcs), vol. 2021, pp. 31:1–31:19. https://doi.org/10.4230/LIPIcs.ITCS.2021.31.Hrushovski E., Ouaknine J., Pouly A. & Worrell J. (2023) On Strongest Algebraic Program Invariants. Journal of the ACM, vol. 70, pp. 1–22. https://doi.org/10.1145/3614319.Hrushovski E., Ouaknine J., Pouly A. & Worrell J. (2023) On Strongest Algebraic Program Invariants. Journal of the ACM, vol. 70, pp. 1–22. https://doi.org/10.1145/3614319.Raith F., Blecha C., Nagel T., Parisio F., Kolditz O., Günther F., Stommel M. & Scheuermann G. (2019) Tensor Field Visualization using Fiber Surfaces of Invariant Space, IEEE Transactions on Visualization and Computer Graphics, vol. 25, pp. 1122–1131. https://doi.org/10.1109/TVCG.2018.2864846.Raith F., Blecha C., Nagel T., Parisio F., Kolditz O., Günther F., Stommel M. & Scheuermann G. (2019) Tensor Field Visualization using Fiber Surfaces of Invariant Space, IEEE Transactions on Visualization and Computer Graphics, vol. 25, pp. 1122–1131. https://doi.org/10.1109/TVCG.2018.2864846.Hillar C.J. & Lim L.H. (2013). Most tensor problems are NP-hard. Journal of the ACM (JACM), 60(6), pp. 1–39.Hillar C.J. & Lim L.H. (2013). Most tensor problems are NP-hard. Journal of the ACM (JACM), 60(6), pp. 1–39.

The authors declare that there are no conflicts of interest present.