Science China Mathematics

, Volume 61, Issue 4, pp 593–626 | Cite as

Holographic software for quantum networks

Open Access
Articles

Abstract

We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.

Keywords

quantum information picture language string Fourier transform Clifford gate multipartite entanglement quantum protocol 

MSC(2010)

81P45 94A15 90B18 94A05 94A17 

Notes

Acknowledgements

This work was supported by the Templeton Religion Trust (Grant Nos. TRT0080 and TRT0159). The authors are grateful for hospitality at the Research Institute for Mathematics (FIM) of the ETH-Zurich, at the Max Planck Institute for Mathematics in Bonn, at the Hausdorff Institute for Mathematics in Bonn, at the Isaac Newton Mathematical Institute in Cambridge, UK, and at the Mathematical Research Institute Oberwolfach, where they did part of this work. The authors thank Erwin Engeler, Klaus Hepp, Daniel Loss, Renato Renner, and Matthias Troyer for helpful discussions. The authors are also grateful to Bob Coecke for sharing a copy of [15], before its publication.

References

  1. 1.
    Aaronson S, Gottesman D. Improved simulation of stabilizer circuits. Phys Rev Lett, 2003, 91: 147902CrossRefGoogle Scholar
  2. 2.
    Abramsky S, Coecke B. A categorical semantics of quantum protocols. In: LICS 2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science. New York: IEEE, 2004, 415–425CrossRefGoogle Scholar
  3. 3.
    Aspelmeyer M, Jennewein T, Pfennigbauer M, et al. Long-distance quantum communication with entangled photons using satellites. In: IEEE Journal of Selected Topics in Quantum Electronics, vol. 9. New York: IEEE, 2003, 1541–1551CrossRefGoogle Scholar
  4. 4.
    Atiyah M F. Topological quantum field theories. Publ Math Inst Hautes Études Sci, 1988, 68: 175–186MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barenco A, Bennett C H, Cleve R, et al. Elementary gates for quantum computation. Phys Rev A (3), 1995, 52: 3457–3467CrossRefGoogle Scholar
  6. 6.
    Baxter R. Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I, II, III. I: Ann Phys (8), 1973, 76: 1–24; II: Ann Phys (8), 1973, 76: 25–47; III: Ann Phys (8), 1973, 76: 48–71MATHGoogle Scholar
  7. 7.
    Bennett C H, Brassard G, Crépeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels. Phys Rev Lett, 1993, 70: 1895–1899MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Biamonte J, Clark S, Jaksch D. Categorical tensor network states. AIP Advances, 2011, 1: 042172CrossRefGoogle Scholar
  9. 9.
    Bombin H, Martin-Delgado M A. Topological computation without braiding. Phys Rev Lett, 2007, 98: 160502MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bose S, Vedral V, Knight P L. Multiparticle generalization of entanglement swapping. Phys Rev A (3), 1998, 57: 822–829CrossRefGoogle Scholar
  11. 11.
    Bremner M, Dawson C, Dodd J, et al. Practical scheme for quantum computation with any two-qubit entangling gate. Phys Rev Lett, 2002, 89: 247902CrossRefGoogle Scholar
  12. 12.
    Brylinski J L, BrylinskiR. Universal Quantum Gates. Mathematics of Quantum Computation. Boca Raton: Chapman & Hall/CRC, 2002CrossRefMATHGoogle Scholar
  13. 13.
    Buerschaper O, Mombelli J, Christandl M, et al. A hierarchy of topological tensor network states. J Math Phys, 2013, 54: 012201MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Campbell E, Anwar H, Browne D. Magic-state distillation in all prime dimensions using quantum Reed-Muller codes. Phys Rev X, 2012, 2: 041021Google Scholar
  15. 15.
    Coecke B, Kissinger A. Picturing Quantum Processes: A First Course in Quantum Theory and Pictorial Reasoning. Cambridge: Cambridge University Press, 2017CrossRefMATHGoogle Scholar
  16. 16.
    Deutsch D. Quantum computational networks. Proc R Soc Lond Ser A Math Phys Eng Sci, 1989, 425: 73–90MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Eisert J, Jacobs K, Papadopoulos P, et al. Optimal local implementation of nonlocal quantum gates. Phys Rev A (3), 2000, 62: 052317CrossRefGoogle Scholar
  18. 18.
    Farinholt J M. An ideal characterization of the Clifford operators. J Phys A, 2014, 47: 305303MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fateev V, Zamolodchikov A B. Self-dual solutions of the star-triangle relations in ZN-models. Phys Lett A, 1982, 92: 37–39MathSciNetCrossRefGoogle Scholar
  20. 20.
    Feynman R. Simulating physics with computers. Internat J Theoret Phys, 1982, 21: 467–488MathSciNetCrossRefGoogle Scholar
  21. 21.
    Freedman M H, Kitaev A, Larsen M J, et al. Topological quantum computation. Bull Amer Math Soc (NS), 2002, 40: 31–38MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Freedman M H, Kitaev A, Wang Z. Simulation of topological field theories by quantum computers. Commun Math Phys, 2002, 227: 587–603MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fröhlich J. New super-selection sectors (‘Soliton-States’) in two-dimensional Bose quantum field models. Comm Math Phys, 1976, 47: 269–310MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fröhlich J. Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Link. Non-Perturbative Quantum Field Theory. New York: Plenum Press, 1988Google Scholar
  25. 25.
    Gottesman D. Stabilizer codes and quantum error correction. Thesis. ArXiv:9705.052, 1997Google Scholar
  26. 26.
    Gottesman D. Theory of fault-tolerant quantum computation. Phys Rev A (3), 1998, 57: 127–137CrossRefGoogle Scholar
  27. 27.
    Gottesman D. The Heisenberg representation of quantum computers. Talk at International Conference on Group Theoretic Methods in Physics. ArXiv:9807.006, 1998Google Scholar
  28. 28.
    Gottesman D, Chuang I L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 1999, 402: 390–393CrossRefGoogle Scholar
  29. 29.
    Greenberger D M, Horne M A, Zeilinger A. Going Beyond Bell’s Theorem. Bell’s Theorem, Quantum Theory, and Conceptions of the Universe. Fundamental Theories of Physics, vol. 37. Heidelberg: Springer, 1989Google Scholar
  30. 30.
    Hu S, Cui W-X, Wang D-Y, et al. Teleportation of a Toffoli gate among distant solid-state qubits with quantum dots embedded in optical microcavities. Nature, 2015, 5: 11321Google Scholar
  31. 31.
    Huelga S F, Vaccaro J A, Che es A, et al. Quantum remote control: teleportation of unitary operations. Phys Rev A (3), 2001, 63: 042303CrossRefMATHGoogle Scholar
  32. 32.
    Hutter A, Loss D. Quantum computing with parafermions. Phys Rev B, 2016, 93: 125105CrossRefGoogle Scholar
  33. 33.
    Jaffe A, Janssens B. Characterization of re ection positivity. Comm Math Phys, 2016, 346: 1021–1050MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Jaffe A, Liu Z. Planar para algebras, re ection positivity. Comm Math Phys, 2017, 352: 95–133MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Jaffe A. Liu Z. A mathematical picture language program. Proc Natl Acad Sci, doi:10.1073/pnas.1710707114, arXiv: 1708.02604, 2017Google Scholar
  36. 36.
    Jaffe A, Liu Z, Wozniakowski A. Qudit isotopy. ArXiv:1602.02671, 2016Google Scholar
  37. 37.
    Jaffe A, Liu Z, Wozniakowski A. Compressed teleportation. ArXiv:1605.00321, 2016Google Scholar
  38. 38.
    Jaffe A, Liu Z, Wozniakowski A. Constructive simulation and topological design of protocols. New J Phys, 2017, 19: 063016CrossRefGoogle Scholar
  39. 39.
    Jaffe A, Pedrocchi F L. Re ection positivity for parafermions. Comm Math Phys, 2015, 337: 455–472MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Jennings D, Brockt C, Haegeman J, et al. Continuum tensor network field states, path integral representations and spatial symmetries. New J Phys, 2015, 17: 063039CrossRefGoogle Scholar
  41. 41.
    Jiang C, Liu Z, Wu J. Noncommutative uncertainty principles. J Funct Anal, 2016, 270: 264–311MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Jiang C, Liu Z, Wu J. Block maps and Fourier analysis. ArXiv:1706.03551, 2017Google Scholar
  43. 43.
    Jones V F R. Index for subfactors. Invent Math, 1983, 72: 1–25MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Jones V F R. A polynomial invariant for knots via von Neumann algebras. Bull Amer Math Soc, 1985, 12: 103–111MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Jones V F R. Hecke algebra representations of braid groups and link polynomials. Ann of Math (2), 1987, 126: 335–388MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Jones V F R. Baxterization. Internat J Modern Phys A, 1991, 6: 2035–2043MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Jones V F R. Planar algebras, I. New Zealand J Math, arXiv:math/9909027, 1998Google Scholar
  48. 48.
    Kauffman L, Lomonaco J S. Comparing quantum entanglement and topological entanglement. New J Phys, 2002, 4: 1–73MathSciNetCrossRefGoogle Scholar
  49. 49.
    Kauffman L, Lomonaco J S. Braiding operators are universal quantum gates. New J Phys, 2004, 6: 1–134MathSciNetCrossRefGoogle Scholar
  50. 50.
    Kimble H J. The quantum internet. Nature, 2008, 453: 1023–1030CrossRefGoogle Scholar
  51. 51.
    Kitaev A. Fault-tolerant quantum computation by anyons. Ann Phys (8), 2003, 303: 2–30MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Lafont Y. Towards an algebraic theory of Boolean circuits. J Pure Appl Algebra, 2003, 184: 257–310MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Levin M, Wen X-G. String-net condensation: A physical mechanism for topological phases. Phys Rev B, 2005, 71: 045110CrossRefGoogle Scholar
  54. 54.
    Liu Z, Jaffe A,Wozniakowski A. Quon 3D language for quantum information. Proc Natl Acad Sci, 2017, 114: 2497–2502MathSciNetCrossRefGoogle Scholar
  55. 55.
    Liu Z, Wang S, Wu J. Young’s inequality for locally compact quantum groups. J Operator Theory, in press, arX-iv:1611.04630, 2016Google Scholar
  56. 56.
    Luo S, Wang A M. Remote implementations of partially unknown quantum operations and its entanglement costs. Http://arxiv.org/pdf/1301.5866.pdf, 2013Google Scholar
  57. 57.
    Ma X-S, Herbst T, Scheidl T, et al. Quantum teleportation over 143 kilometres using active feed-forward. Nature, 2012, 489: 269–273CrossRefGoogle Scholar
  58. 58.
    Manin Y. Computable and Uncomputable (in Russian). Moscow: Sovetskoye Radio, 1980Google Scholar
  59. 59.
    Manin Y. Classical computing, quantum computing, and Shor’s factoring algorithm. Astérisque, 2000, 266: 375–404MathSciNetMATHGoogle Scholar
  60. 60.
    Nayak C, Simon S H, Stern A, et al. Non-Abelian anyons and topological quantum computation. Rev Modern Phys, 2008, 80: 1083–1159MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Nielsen M A, Chuang I L. Programmable quantum gate arrays. Phys Rev Lett, 1997, 79: 321–324MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2010CrossRefMATHGoogle Scholar
  63. 63.
    Ocneanu A. Quantized groups, string algebras and Galois theory for algebras. In: Operator Algebras and Applications, vol. 2, London Mathematical Society Lecture Note Series, vol. 136. Cambridge: Cambridge University Press, 1988, 119–172Google Scholar
  64. 64.
    Ogburn R W, Preskill J. Topological quantum computation quantum computing and quantum communications. In: Lecture Notes in Computer Science, vol. 1509. Berlin-Heidelberg: Springer, 1999, 341–356CrossRefMATHGoogle Scholar
  65. 65.
    Pan J-W. Quantum science satellite. Chinese J Space Sci, 2014, 34: 547–549Google Scholar
  66. 66.
    Penrose R. Application of negative dimension tensors. In: Combinatorial Mathematics and Its Applications. New York: Academic Press, 1971, 221–244Google Scholar
  67. 67.
    Ren J-G, Xu P, Yong H L, et al. Ground-to-satellite quantum teleportation. Nature, 2017, 549: 70–73CrossRefGoogle Scholar
  68. 68.
    Reshetikhin N, Turaev V. Invariants of 3-manifolds via link polynomials and quantum groups. Invent Math, 1991, 103: 547–597MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Reznik B, Aharonov Y, Groisman B. Remote operations and interactions for systems of arbitrary-dimensional hilbert space: state-operator approach. Phys Rev A (3), 2002, 65: 032312CrossRefGoogle Scholar
  70. 70.
    Schliemann J, Ignacio Cirac J, Kuś M, et al. Quantum correlations in two-fermion systems. Phys Rev A (3), 2001, 64: 022303CrossRefGoogle Scholar
  71. 71.
    Schliemann J, Loss D, MacDonald A H. Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots. Phys Rev B, 2001, 63: 085311CrossRefGoogle Scholar
  72. 72.
    Sørensen A, Mølmer K. Error-free quantum communication through noisy channels. Phys Rev A (3), 1998, 58: 2745–2749CrossRefGoogle Scholar
  73. 73.
    van Loock P, Braunstein S L. Multipartite entanglement for continuous variables: A quantum teleportation network. Phys Rev Lett, 2000, 84: 3482–3485CrossRefGoogle Scholar
  74. 74.
    Van Meter R. Quantum Networking. Hoboken: John Wiley & Sons, 2014CrossRefMATHGoogle Scholar
  75. 75.
    Vidal G. Efficient classical simulation of slightly entangled quantum computations. Phys Rev A (3), 2004, 70: 052328CrossRefGoogle Scholar
  76. 76.
    Witten E. Topological quantum field theory. Comm Math Phys, 1988, 117: 353–386MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Yang C N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys Rev Lett, 1967, 19: 1312–1315MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Yin J, Cao Y, Li Y H, et al. Satellite-based entanglement distribution over 1200 kilometers. Science, 2017, 356: 1140–1144CrossRefGoogle Scholar
  79. 79.
    Yin J, Ren J-G, Lu H, et al. Quantum teleportation and entanglement distribution over 100-kilometre free-space channels. Nature, 2012, 488: 185–188CrossRefGoogle Scholar
  80. 80.
    Yu L, Griffiths R B, Cohen S M. Efficient implementation of bipartite nonlocal unitary gates using prior entanglement and classical communication. Phys Rev A (3), 2010, 81: 062315CrossRefGoogle Scholar
  81. 81.
    Zhao N B, Wang A M. Hybrid protocol of remote implementations of quantum operations. Phys Rev A (3), 2007, 76: 062317CrossRefGoogle Scholar
  82. 82.
    Zhou X, Leung D W, Chuang I L. Methodology for quantum logic gate construction. Phys Rev A (3), 2000, 62: 052316CrossRefGoogle Scholar
  83. 83.
    Zukowski M, Zeilinger A, Horne M A, et al. ‘Event-ready-detectors’ Bell experiment via entanglement swapping. Phys Rev Lett, 1993, 71: 4287–4290CrossRefGoogle Scholar

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© The Authors 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

This article is published with open access at Springerlink.com, corrected publication 03/2018

The original article has been corrected.

Authors and Affiliations

  1. 1.Departments of Mathematics and PhysicsHarvard UniversityCambridgeUSA

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