Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Quantum Information Processing
Erschienen in: Quantum Information Processing 9/2020

01.08.2020

Learning algebraic models of quantum entanglement

verfasst von: Hamza Jaffali, Luke Oeding

Erschienen in: Quantum Information Processing | Ausgabe 9/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We review supervised learning and deep neural network design for learning membership on algebraic varieties. We demonstrate that these trained artificial neural networks can predict the entanglement type for quantum states. We give examples for detecting degenerate states, as well as border rank classification for up to 5 binary qubits and 3 qutrits (ternary qubits).
Vorheriger Artikel Quantum algorithm for Help-Training semi-supervised support vector machine
Nächster Artikel Adiabatic speedup in cutting a spin chain by pulse control in a laboratory frame

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Fußnoten
1
By “general” we mean avoiding a measure-zero set of possible counterexamples.
 
2
The justification of the term “high probability” is the following. For real numbers, the set of degenerate tensors has codimension at least 1 in the ambient space \(\mathbb {R}^{d^{n}}\) and thus it has measure zero. Thus, the probability is zero that a tensor chosen randomly from \(\mathbb {R}^{d^{n}}\) lands in a measure zero set. When floating point precision is used, one expects that these continuous concepts are still well-approximated in the discrete case, even despite the fact that a non-empty subset (the exceptions) never has measure zero in this case.
 
Literatur
1.
Abo, H., Ottaviani, G., Peterson, C.: Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc. 361(2), 767–792 (2009)MathSciNetMATH
2.
Anthony, M., Bartlett, P.L.: Neural network learning: Theoretical foundations. Cambridge University Press, New York (2009)MATH
3.
Barron, A., Barron, R.: Statistical learning networks: A unifying view, In: Proceedings of the 20th Symposium Computer Science and Statistics (1988)
4.
Baur, K., Draisma, J., de Graaf, W.A.: Secant dimensions of minimal orbits: computations and conjectures. Exp. Math. 16(2), 239–250 (2007)MathSciNetMATH
5.
Beach, M.J.S., Vlugt, I.D., Golubeva, A., Huembeli, P., Kulchytskyy, B., Luo, X., Melko, R.G., Merali, E., Torlai, G.: QuCumber: wavefunction reconstruction with neural networks. SciPost Phys. 7, 9 (2019)ADS
6.
Behrman, E.C., Steck, J.E.: Dynamic learning of pairwise and three-way entanglement, In: 2011 Third World Congress on Nature and Biologically Inspired Computing, pp. 99–104 (2011)
7.
Belkin, M., Hsu, D., Ma, S., Mandal, S.: Reconciling modern machine-learning practice and the classical bias-variance trade-off. PNAS 116(32), 15849–15854 (2019)MathSciNetMATH
8.
Berkovits, R.: Extracting many-particle entanglement entropy from observables using supervised machine learning. Phys. Rev. B 98(24), 241411 (2018)ADS
9.
Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature 549(7671), 195–202 (2017)ADS
10.
Bini, D., Capovani, M., Romani, F., Lotti, G.: \(O(n^{2.7799})\) complexity for \(n\times n\) approximate matrix multiplication. Inform. Process. Lett. 8(5), 234–235 (1979)MathSciNetMATH
11.
Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press Inc, USA (1995)MATH
12.
Borne, P., Benrejeb, M., Haggège, J.: Les réeseaux de neurones: Presentation et Applications, vol. 15, Editions OPHRYS, (2007)
13.
Brambilla, M.C., Ottaviani, G.: On the Alexander–Hirschowitz theorem. J. Pure Appl. Algebra 212(5), 1229–1251 (2008)MathSciNetMATH
14.
Breiding, P., Kališnik, S., Sturmfels, B., Weinstein, M.: Learning algebraic varieties from samples. Revista Mat. Complut. 31(3), 545–593 (2018)MathSciNetMATH
15.
Brylinski, J.-L.: Algebraic measures of entanglement, In: Mathematics of Quantum Computation, pp. 19–40 (2002)
16.
Carlini, E., Grieve, N., Oeding, L.: Four lectures on secant varieties, In: Connections Between Algebra, Combinatorics, and Geometry, pp. 101–146 (2014)
17.
Che, M., Qi, L., Wei, Y., Zhang, G.: Geometric measures of entanglement in multipartite pure states via complex-valued neural networks. Neurocomputing 313, 25–38 (2018)
18.
Chen, L., Đoković, D.Ž.D.Ž.: Proof of the Gour-Wallach conjecture. Phys. Rev. A 88(4), 042307 (2013) ADS
19.
Cheng, X., Khomtchouk, B., Matloff, N., Mohanty, P.: Polynomial regression as an alternative to neural nets. arXiv:​1806.​06850 (2018)
20.
Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Math. Anal. 35(4), 1265–1287 (2014)MathSciNetMATH
21.
22.
23.
Cirrincione, G., Cirrincione, M.: Neural-Based Orthogonal Data Fitting: The EXIN Neural Networks, vol. 66. Wiley, New York (2011)MATH
24.
Csáji, B.C., et al.: Approximation with artificial neural networks. Fac. Sci. Etvs Lornd Univ. Hung. 24(48), 7 (2001)
25.
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control, Signals, and Systems 5(4), 455–455 (1989)MathSciNetMATH
26.
27.
de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008)MathSciNetMATH
28.
29.
Dreyfus, G., Martinez, J.-M., Samuelides, M., Gordon, M.B., Badran, F., Thiria, S., Hérault, L.: Réseaux de neurones-méthodologie et applications, Eyrolles (2002)
30.
Duda, R.O., Hart, P.E., et al.: Pattern Classification and Scene Analysis, vol. 3. Wiley, New York (1973)MATH
31.
Dufresne, E., Edwards, P., Harrington, H., Hauenstein, J.: Sampling real algebraic varieties for topological data analysis, In: 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pp. 1531–1536 (2019)
32.
Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)ADSMathSciNet
33.
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory and Applications. Birkhäuser, Boston (1994)MATH
34.
Gibson, G.J., Cowan, C.F.N.: On the decision regions of multilayer perceptrons. Proc. IEEE 78(10), 1590–1594 (1990)
35.
36.
Govender, L.: Determination of quantum entanglement concurrence using multilayer perceptron neural networks., Ph.D. Thesis, (2017)
37.
Gray, J., Banchi, L., Bayat, A., Bose, S.: Machine-learning-assisted many-body entanglement mea- surement. Phys. Rev. Lett. 121(15), 150503 (2018)ADS
38.
Hanin, B.: Universal function approximation by deep neural nets with bounded width and ReLU activations. Mathematics 7(10), 992 (2019)
39.
Hassoun, M.H., et al.: Fundamentals of Artificial Neural Networks. MIT Press, Cambridge (1995)MATH
40.
Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, USA (1999)MATH
41.
Hernández-Mederos, V., Estrada-Sarlabous, J.: Sampling points on regular parametric curves with control of their distribution. Comput. Aided Geom. Design 20(6), 363–382 (2003)MathSciNetMATH
42.
Holweck, F., Luque, J.-G., Thibon, J.-Y.: Geometric descriptions of entangled states by auxiliary varieties. J. Math. Phys. 53(10), 102203 (2012)ADSMathSciNetMATH
43.
Holweck, F., Luque, J.-G., Thibon, J.-Y.: Entanglement of four-qubit systems: A geometric atlas with polynomial compass II (the tame world). J. Math. Phys. 58(2), 022201 (2017)ADSMathSciNetMATH
44.
Holweck, F., Luque, J.-G., Thibon, J.-Y.: Entanglement of four qubit systems: A geometric atlas with polynomial compass I (the finite world). J. Math. Phys. 55(1), 12202 (2014)MathSciNetMATH
45.
46.
Holweck, F., Jaffali, H.: Three-qutrit entanglement and simple singularities. J. Phys. A 49(46), 465301 (2016)ADSMathSciNetMATH
47.
Holweck, F., Jaffali, H., Levay, P., Luque, J.-G.: Maximally entangled symmetric states, (in preparation) (2019)
48.
Holweck, F., Jaffali, H., Nounouh, I.: Grover’s algorithm and the secant varieties. Quantum Inf. Process. 15(11), 4391–4413 (2016)ADSMathSciNetMATH
49.
Holweck, F., Luque, J.-G., Planat, M.: Singularity of type d4 arising from four qubit systems. J. Phys. A 47(13), 135301 (2014)ADSMathSciNetMATH
50.
Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251–257 (1991)MathSciNet
51.
Huggins, P., Sturmfels, B., Yu, J., Yuster, D.: The hyperdeterminant and triangulations of the 4-cube. Math. Comp. 77(263), 1653–1679 (2008)ADSMathSciNetMATH
52.
Ivakhnenko, A.G.: Polynomial theory of complex systems. IEEE Trans. Syst. Man Cybern. 4, 364–378 (1971)MathSciNet
53.
Jaffali, H., Holweck, F.: Quantum entanglement involved in Grover’s and Shor’s algorithms: the four-qubit case. Quantum Inf. Process. 18(5), 133 (2019)ADSMathSciNet
54.
55.
Kerenidis, I., Landman, J., Luongo, A., Prakash, A.: q-means: A quantum algorithm for unsupervised machine learning. In: Advances in Neural Information Processing Systems, pp. 4136–4146 (2019)
56.
Khanna, T.: Foundations of Neural Networks. Addison Wesley, Reading (1990)MATH
57.
Kileel, J., Trager, M., Bruna, J.: On the expressive power of deep polynomial neural networks. In: Advances in Neural Information Processing Systems, pp. 10310–10319 (2019)
58.
Kung, S.Y., Diamantaras, K., Mao, W.D., Taur, J.S.: Generalized perceptron networks with non-linear discriminant functions. In: Neural Networks, 245–279 (1992)
59.
Landsberg, J.M.: Tensors: Geometry and Applications, Graduate Studies in Mathematics, vol. 128. American Mathematical Society, Providence (2012)
60.
Landsberg, J.M.: Geometry and Complexity Theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, New York (2017)
61.
Lanyon, B.P., Langford, N.K.: Experimentally generating and tuning robust entanglement between photonic qubits. New J. Phys. 11(1), 013008 (2009)ADS
62.
Li, H.-X., Lee, E.S.: Interpolation functions of feedforward neural networks. Comput. Math. Appl. 46(12), 1861–1874 (2003)MathSciNetMATH
63.
Lippmann, R.P.: An introduction to computing with neural nets. ACM Sigarch Comput. Archit. News 16(1), 7–25 (1988)
64.
Llanas, B., Sainz, F.J.: Constructive approximate interpolation by neural networks. J. Comput. Appl. Math. 188(2), 283–308 (2006)ADSMathSciNetMATH
65.
Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum principal component analysis. Nature Phys. 10(9), 631–633 (2014)ADS
66.
Lu, S., Huang, S., Li, K., Li, J., Chen, J., Lu, D., Ji, Z., Shen, Y., Zhou, D., Zeng, B.: Separability-entanglement classifier via machine learning. Phys. Rev. A 98(1), 012315 (2018)ADS
67.
Z. Lu, H. Pu, F. Wang, Z. Hu, and L. Wang, The expressive power of neural networks: A view from the width, In: Neural Information Processing Systems, pp. 6231–6239 (2017)
68.
Luque, J.-G., Thibon, J.-Y.: Polynomial invariants of four qubits. Phys. Rev. A 67(4), 42303 (2003)ADSMathSciNet
69.
70.
Ma, Y.-C., Yung, M.-H.: Transforming Bell’s inequalities into state classifiers with machine learning. npj Quantum Inf. 4(1), 34 (2018)ADS
71.
McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biol. 52(4), 99–115 (1990)MATH
72.
Miyake, A.: Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A 67, 012108 (2003)ADSMathSciNet
73.
Miyake, A., Verstraete, F.: Multipartite entanglement in \(2 \times 2\times n\) quantum systems. Phys. Rev. A 69(1), 12101 (2004)ADSMathSciNet
74.
Miyake, A., Wadati, M.: Multipartite entanglement and hyperdeterminants. Quantum Inf. Comput. 2(7), 540–555 (2002)MathSciNetMATH
75.
Nautrup, H.P., Delfosse, N., Dunjko, V., Briegel, H.J., Friis, N.: Optimizing quantum error correction codes with reinforcement learning. Quantum 3, 215 (2019)
76.
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th anniversary edition, 10th edn. Cambridge University Press, New York (2011)MATH
77.
Oh, S.-K., Pedrycz, W., Park, B.-J.: Polynomial neural networks architecture: analysis and design. Comput. Electr. Eng. 29(6), 703–725 (2003)
78.
Ong, H.C., Kang, L.C., Wui, Y.Y.: Non linear approximations using multi-layered perceptions and polynomial regressions. In: Proceedings of the 2nd IMT-GT Regional Conference on Mathematics, Statistics and Applications, Universiti Sains Malaysia, Penang, June 13-15 (2006)
79.
Osterloh, A., Siewert, J.: Entanglement monotones and maximally entangled states in multipartite qubit systems. Int. J. Quantum Inf. 4(3), 531–540 (2006)MATH
80.
Pagani, L., Scott, P.J.: Curvature based sampling of curves and surfaces. Comput. Aided Geom. Des. 59, 32–48 (2018)MathSciNetMATH
81.
82.
Raturi, R.: Large data analysis via interpolation of functions: Interpolating polynomials vs artificial neural networks. Am. J. Int. Syst. 8(1), 6–11 (2018)
83.
Rebentrost, P., Mohseni, M., Lloyd, S.: Quantum support vector machine for big data classification. Phys. Rev. Lett. 113(13), 130503 (2014)ADS
84.
Schuld, M., Sinayskiy, I., Petruccione, F.: The quest for a quantum neural network. Quantum Inf. Process. 13(11), 2567–2586 (2014)ADSMathSciNetMATH
85.
Schuld, M., Sinayskiy, I., Petruccione, F.: An introduction to quantum machine learning. Contemp. Phys. 56(2), 172–185 (2015)ADSMATH
86.
Segre, C.: Sulle corrispondenze quadrilineari tra forme di 1.a specie e su alcune loro rappresentazioni spaziali, Annali Mat. Pura ed Applicata 29, 105–140 (1920)MATH
87.
Sheng, Y.-B., Zhou, L.: Distributed secure quantum machine learning. Sci. Bull. 62(14), 1025–1029 (2017)
88.
Shin, Y., Ghosh, J.: Approximation of multivariate functions using ridge polynomial networks, In: Proceedings 1992 IJCNN International Joint Conference on Neural Networks, pp. 380–385 (1992)
89.
Shin, Y., Ghosh, J.: Ridge polynomial networks. IEEE Trans. Neural Netw. 6(3), 610–622 (1995)
90.
Specht, D.F.: Generation of polynomial discriminant functions for pattern recognition. IEEE Trans. Electron. Comput. 16(3), 308–319 (1967)MATH
91.
Sutskever, I., Martens, J., Dahl, G.E., Hinton, G.E.: On the importance of initialization and momentum in deep learning, In: Proceedings of the 30th International Conference on Machine Learning, pp. 1139–1147 (2013)
92.
Vemuri, V.R.: Artificial Neural Networks: Concepts and Control Applications. IEEE Computer Society Press, Los Alamitos (1992)MATH
93.
Verstraete, F., Dehaene, J., Moor, B.D., Verschelde, H.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65(5), 52112 (2002)ADSMathSciNet
94.
Vinberg, É.B., Élašvili, A.G.: A classification of the three-vectors of nine-dimensional space. Trudy Sem. Vektor. Tenzor. Anal. 18, 197–233 (1978)MathSciNet
95.
96.
Wang, L., Alkon, D.L.: Artificial Neural Networks: Oscillations, Chaos, and Sequence Processing, Neural Networks Technology Series. IEEE Computer Society Press, Los Alamitos (1993)MATH
97.
Weyman, J., Zelevinsky, A.: Singularities of hyperdeterminants. Ann. Inst. Fourier (Grenoble) 46(3), 591–644 (1996) MathSciNetMATH
98.
Widrow, B., Lehr, M.A.: 30 years of adaptive neural networks: perceptron, madaline, and backprop- agation. Proc. IEEE 78(9), 1415–1442 (1990)
99.
Wiebe, N., Braun, D., Lloyd, S.: Quantum algorithm for data fitting. Phys. Rev. Lett. 109, 050505 (2012)ADS
100.
Wiebe, N., Kapoor, A., Svore, K.M.: Quantum deep learning. Quantum Info. Comput. 16(7–8), 541–587 (2016)MathSciNet
101.
Wišniewska, J., Sawerwain, M.: Detecting entanglement in quantum systems with artificial neural network, In: Intelligent Information and Database Systems, pp. 358–367 (2015)
102.
Witten, I.H., Frank, E., Hall, M.A., Pal, C.J.: Data Mining: Practical Machine Learning Tools and Techniques, 4th edn. Morgan Kaufmann Publishers Inc., San Francisco (2016)
103.
Wossnig, L., Severini, S.: Quantum Machine Learning: Challenges and Opportunities. Bulletin of the American Physical Society, (2019)
104.
Xu, W., Zhao, X., Long, G.: Efficient generation of multi-photon w states by joint-measurement. Prog. Nat. Sci. 18(1), 119–122 (2008)MathSciNet
105.
Yang, M., Ren, C.-L., Ma, Y.-C., Xiao, Y., Ye, X.-J., Song, L.-L., Xu, J.-S., Yung, M.-H., Li, C.-F., Guo, G.-C.: Experimental simultaneous learning of multiple nonclassical correlations. Phys. Rev. Lett. 123(19), 190401 (2019)ADS
Metadaten
Titel
Learning algebraic models of quantum entanglement
verfasst von
Hamza Jaffali
Luke Oeding
Publikationsdatum
01.08.2020
Verlag
Springer US
Erschienen in
Quantum Information Processing / Ausgabe 9/2020
Print ISSN: 1570-0755
Elektronische ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-020-02785-4

Weitere Artikel der Ausgabe 9/2020

Quantum Information Processing 9/2020 Zur Ausgabe

OriginalPaper

Quantum steering and quantum coherence in XY model with Dzyaloshinskii–Moriya interaction

OriginalPaper

Quantum algorithm for Help-Training semi-supervised support vector machine

OriginalPaper

Advanced exact synthesis of Clifford+T circuits

OriginalPaper

Two-acoustic-cavity interaction mediated by superconducting artificial atoms

OriginalPaper

Novel quantum image compression and encryption algorithm based on DQWT and 3D hyper-chaotic Henon map

OriginalPaper

Vacuum-based quantum random number generator using multi-mode coherent states

Neuer Inhalt

    Bildnachweise