Computing spin networks. (English) Zbl 1072.81013
The paper proposes a general setting for the quantum structure of quantum information. The framework is a non-Boolean generalization of the quantum circuit model based on a combinatorial approach to spin networks. After a careful description of computational Hilbert spaces, gates are defined as unitary transformations on these spaces. If the spin network quantum circuit is defined, one deals with semi-classical and SU(2) state sum models on the one hand, and spin network together with topological quantum computation, on the other hand.
Reviewer: Guy Jumarie (Montréal)
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| [1] | DiVincenzo, D. P.; Bennett, C., Nature, 404, 247 (2000) · Zbl 1369.81023 |
| [2] | R. Jozsa, Entanglement and quantum computation. Available from preprint: <quant-ph/9707034>; R. Jozsa, Entanglement and quantum computation. Available from preprint: <quant-ph/9707034> |
| [3] | Kitaev, A., Ann. Phys., 303, 2 (2003) · Zbl 1012.81006 |
| [4] | Freedman, M. H.; Kitaev, A.; Larsen, M.; Wang, Z., Bull. Am. Math. Soc., 40, 31 (2002) |
| [5] | Jones, J. A.; Vedral, V.; Ekert, A.; Castagnoli, G., Nature, 403, 869 (2000) |
| [6] | S. Lloyd, Quantum computation with Abelian anyons. Available from preprint: <quant-ph/0004010>; S. Lloyd, Quantum computation with Abelian anyons. Available from preprint: <quant-ph/0004010> |
| [7] | Dennis, E.; Kitaev, A. Yu.; Landahl, A.; Preskill, J., J. Math. Phys., 43, 4452 (2002) · Zbl 1060.94045 |
| [8] | S.B. Bravyi, A.Yu. Kitaev, Fermionic quantum computation. Available from preprint: <quant-ph/0003137>; S.B. Bravyi, A.Yu. Kitaev, Fermionic quantum computation. Available from preprint: <quant-ph/0003137> |
| [9] | Zanardi, P.; Rasetti, M., Phys. Lett. A, 264, 94 (1999) · Zbl 0949.81009 |
| [10] | Pachos, J.; Zanardi, P.; Rasetti, M., Phys. Rev. A, 61, 010305(R) (2000) |
| [11] | Marzuoli, A.; Rasetti, M., Phys. Lett. A, 306, 79 (2002) · Zbl 1005.81015 |
| [12] | A.P. Yutsis, I.B. Levinson, V.V. Vanagas, The Mathematical Apparatus of the Theory of Angular Momentum, Israel Program for Sci. Transl. Ltd., Jerusalem, 1962; A.P. Yutsis, I.B. Levinson, V.V. Vanagas, The Mathematical Apparatus of the Theory of Angular Momentum, Israel Program for Sci. Transl. Ltd., Jerusalem, 1962 · Zbl 0111.42704 |
| [13] | L.C. Biedenharn, J.D. Louck, G.-C. Rota (Ed.), The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Applications, vol. 9, Addison-Wesley, Reading, MA, 1981, Topic 9. Physical Interpretation and Asymptotic (Classical) Limits of the Angular Momentum Functions; Topic 12. Coupling of \(N\) Angular Momenta: Recoupling Theory; L.C. Biedenharn, J.D. Louck, G.-C. Rota (Ed.), The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Applications, vol. 9, Addison-Wesley, Reading, MA, 1981, Topic 9. Physical Interpretation and Asymptotic (Classical) Limits of the Angular Momentum Functions; Topic 12. Coupling of \(N\) Angular Momenta: Recoupling Theory |
| [14] | Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K., Quantum theory of angular momentum (1988), World Scientific: World Scientific Singapore |
| [15] | Feynman, R. P., Int. J. Theor. Phys., 21, 467 (1982) |
| [16] | Penrose, R., Angular momentum: an approach to combinatorial space-time, (Bastin, T., Quantum theory and beyond (1971), Cambridge University Press: Cambridge University Press Cambridge), 151 |
| [17] | S. Garnerone, A. Marzuoli, M. Rasetti, in preparation; S. Garnerone, A. Marzuoli, M. Rasetti, in preparation |
| [18] | Ponzano, G.; Regge, T., Semiclassical limit of Racah coefficients, (Bloch, F.; etal., Spectroscopic and group theoretical methods in physics (1968), North-Holland: North-Holland Amsterdam), 1 · Zbl 0201.30804 |
| [19] | Biedenharn, L. C.; Louck, J. D., Angular momentum in quantum physics, theory and applications, (Rota, G.-C., Encyclopedia of mathematics and its applications, vol. 8 (1981), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0474.00023 |
| [20] | Regge, T., Nuovo Cimento, 19, 558 (1961) |
| [21] | Kempe, J.; Bacon, D.; Lidar, D. A.; Whaley, K. B., Phys. Rev. A, 63, 042307 (2001) |
| [22] | Fack, V.; Lievens, S.; Van der Jeugt, J., Comp. Phys. Commun., 119, 99 (1999) |
| [23] | Fack, V.; Lievens, S.; Van der Jeugt, J., Discr. Math., 245, 1 (2002) · Zbl 0993.05064 |
| [24] | Barenco, A.; Bennett, C. H.; Cleve, R.; DiVincenzo, D. P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J.; Weinfurter, H., Phys. Rev. A, 52, 3457 (1995) |
| [25] | Bernstein, E.; Vazirani, U., SIAM J. Comput., 26, 1411 (1997) · Zbl 0895.68042 |
| [26] | Moore, C.; Crutchfield, J. P., Theor. Comput. Sci., 237, 275 (2000) · Zbl 0939.68037 |
| [27] | Aquilanti, V.; Coletti, C., Chem. Phys. Lett., 344, 601 (2001) |
| [28] | Askey, R., Ortogonal polynomials and special functions (1975), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0298.33008 |
| [29] | Marzuoli, A.; Rasetti, M., Int. J. Quantum Infor., 3, 65 (2005) · Zbl 1133.81325 |
| [30] | Lloyd, S., Science, 273, 1073 (1996) · Zbl 1226.81059 |
| [31] | Stanley, R. P., Enumerative combinatorics, vol. 2 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0928.05001 |
| [32] | Kauffman, L. H., Knots and physics (1991), World Scientific: World Scientific Singapore · Zbl 0749.57002 |
| [33] | Ambjorn, J.; Durhuus, B.; Jonsson, T., Quantum geometry (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0993.82500 |
| [34] | Regge, T.; Williams, R. M., J. Math. Phys., 41, 3964 (2000) · Zbl 0984.83016 |
| [35] | Carbone, G.; Carfora, M.; Marzuoli, A., Nucl. Phys. B, 595, 654 (2001) · Zbl 0972.82016 |
| [36] | V. Aquilanti, private communication; V. Aquilanti, private communication |
| [37] | Turaev, V. G.; Viro, O. Ya., Topology, 31, 865 (1992) · Zbl 0779.57009 |
| [38] | Turaev, V. G., Quantum invariants of knots and 3-manifolds (1994), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 0812.57003 |
| [39] | Carlip, S., Quantum gravity in 2+1 dimensions (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0938.83010 |
| [40] | Birmingham, D.; Blau, M.; Rakowski, M.; Thompson, G., Phys. Rep., 209, 129 (1991) |
| [41] | Atiyah, M. F., Publ. Math. Inst. Hautes Etudes Sci., 68, 175 (1989) |
| [42] | Quinn, F., Lectures on axiomatic topological quantum field theories, (Freed, D. S.; etal., Geometry and quantum field theories. Geometry and quantum field theories, IAS/Park City Math. Series, vol. 1 (1995), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0901.18002 |
| [43] | Jones, V., Bull. Am. Math. Soc., 12, 103 (1985) · Zbl 0564.57006 |
| [44] | Witten, E., Commun. Math. Phys., 121, 351 (1989) · Zbl 0667.57005 |
| [45] | Beckman, D.; Gottesman, D.; Kitaev, A. Yu.; Preskill, J., Phys. Rev. D, 65, 065022 (2002) |
| [46] | Jaeger, F.; Vertigen, D.; Welsh, D., Math. Proc. Cambridge Philos. Soc., 108, 35 (1990) · Zbl 0747.57006 |
| [47] | Garey, M. R.; Johnson, D. S., Computers and intractability. A guide to the theory of NP-completeness (1979), W.H. Freeman: W.H. Freeman New York · Zbl 0411.68039 |
| [48] | Di Vincenzo, D. P., Phys. Rev. A, 50, 1015 (1995) |
| [49] | Wilczek, F.; Zee, A., Phys. Rev. Lett., 52, 2111 (1984) |
| [50] | Jackiw, R., (DeWitt, B. S.; Stora, R., Relativity groups and topology, Les Houches 1983 (1984), North-Holland: North-Holland Amsterdam), 154 pp |
| [51] | Nakahara, M., Geometry, topology and physics (1990), IOP Publishing · Zbl 0764.53001 |
| [52] | Preskill, J., Fault-tolerant quantum computation, (Lo, Hoi-Kwong; Popescu, S.; Spiller, T., Introduction to quantum computation and information (1999), World Scientific: World Scientific Singapore) · Zbl 0957.68045 |
| [53] | L.H. Kauffman, S.J. Lomonaco, Braiding operators are universal quantum gates. Available from preprint: <quant-ph/0401090>; L.H. Kauffman, S.J. Lomonaco, Braiding operators are universal quantum gates. Available from preprint: <quant-ph/0401090> |
| [54] | Fuchs, J.; Schweigert, C., Symmetries, Lie algebras and representations (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0923.17001 |
| [55] | Smorodinskii, Ya. A.; Shelepin, L. A., Sov. Phys. Usp., 15, 1 (1972) |
| [56] | On-Line Encyclopedia of Integer Sequences, at http://www.research.att.com/njas/sequences/; On-Line Encyclopedia of Integer Sequences, at http://www.research.att.com/njas/sequences/ · Zbl 1274.11001 |
| [57] | Buckley, F.; Harari, F., Distance in graphs (1990), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0688.05017 |
| [58] | Sleator, D. D.; Tarjan, R. E.; Thurston, W. P., J. Am. Math. Soc., 1, 647 (1988) · Zbl 0653.51017 |
| [59] | Li, M.; Tromp, J.; Zhang, L., J. Theor. Biol., 182, 463 (1996) |
| [60] | Culik, K.; Wood, D., Inform. Process. Lett., 15, 39 (1982) · Zbl 0489.68058 |
| [61] | Pallo, J. M., Inform. Process. Lett., 73, 87 (2000) · Zbl 1014.68112 |
| [62] | Pallo, J. M., Inform. Process. Lett., 87, 173 (2003) · Zbl 1161.68692 |
| [63] | Rogers, R. O.; Dutton, R. D., Congr. Numer., 120, 103 (1996) · Zbl 0901.05041 |
| [64] | Cleary, S.; Tabak, J., Inform. Process. Lett., 88, 251 (2003) · Zbl 1178.68174 |
| [65] | B. DasGupta, X. He, T. Jiang, M. Li, J. Tromp, L. Zhang, in: DIMACS Series in Discr. Math. and Theor. Comp. Science, vol. 55, American Mathematical Society, Providence, RI, 2000, p. 125; B. DasGupta, X. He, T. Jiang, M. Li, J. Tromp, L. Zhang, in: DIMACS Series in Discr. Math. and Theor. Comp. Science, vol. 55, American Mathematical Society, Providence, RI, 2000, p. 125 |
| [66] | Majorana, E., Nuovo Cimento, 9, 43 (1932) · JFM 58.1373.01 |
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