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Topological quantum computation. (English) Zbl 1019.81008
This is a survey paper describing quantum computation based on unitary topological modular functors. The chief advantage of topological computation is physical error correction, a consequence of topological stability. In contrast, other quantum computation models, which repair errors combinatorially, require an extremely low initial error rate.
The theoretical model considered by the authors is obtained by restricting to planar surfaces the Witten-Chern-Simons modular functor with gauge grop $$\text{SU}(2)$$ at fifth roots of unity. Logical qubits are punctured disks with punctures colored by irreducible representations, and the gates (unitary transformations) are the operators of the Hecke algebra representation discovered by Jones. The authors conjecture that such a model could by implemented physically using the braiding and fusion of anyonic excitation in quantum Hall electron liquids with $$\nu={5\over 2}$$.

##### MSC:
 81P68 Quantum computation 57R56 Topological quantum field theories (aspects of differential topology) 81T45 Topological field theories in quantum mechanics 68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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##### References:
 [1] S. Bravyi, A. Kitaev, private communications. [2] D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A 400 (1985), no. 1818, 97 – 117. · Zbl 0900.81019 [3] D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London Ser. A 425 (1989), no. 1868, 73 – 90. · Zbl 0691.68054 [4] Torbjörn Einarsson, Fractional statistics on a torus, Phys. Rev. Lett. 64 (1990), no. 17, 1995 – 1998. · Zbl 1050.81566 [5] Richard P. Feynman, Simulating physics with computers, Internat. J. Theoret. Phys. 21 (1981/82), no. 6-7, 467 – 488. Physics of computation, Part II (Dedham, Mass., 1981). [6] Richard P. Feynman, Quantum mechanical computers, Found. Phys. 16 (1986), no. 6, 507 – 531. [7] M. Freedman, A. Kitaev, and Z. Wang, Simulation of topological field theories by quantum computers, Comm. Math. Phys. 227 (2002), no. 3, 587-603. · Zbl 1014.81006 [8] M. Freedman, M. Larsen, and Z. Wang, A modular functor which is universal for quantum computation, Comm. Math. Phys. 227 (2002), no. 3, 605-622. · Zbl 1012.81007 [9] M. Freedman, M. Larsen, and Z. Wang, The two-eigenvalue problem and density of Jones representation of braid groups, Comm. Math. Phys. 228 (2002), no. 1, 177-199. CMP 2002:14 [10] M.H. Freedman, Quantum computation and the localization of modular functors, Found. Comput. Math. 1 (2001), no. 2, 183-204. · Zbl 1004.57026 [11] Michael H. Freedman, P/NP, and the quantum field computer, Proc. Natl. Acad. Sci. USA 95 (1998), no. 1, 98 – 101. · Zbl 0895.68053 [12] T. Senthil and M.P.A. Fisher, Fractionalization, topological order, and cuprate superconductivity, cond-mat/0008082. [13] S. Girvin, The quantum Hall effect: novel excitations and broken symmetries, in Topological aspects of low dimensional systems, Edited by A. Comtet, T. Jolicoeur, S. Ouvry, and F. David EDP Sci., Les Ulis, 1999. · Zbl 0993.81068 [14] D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. Lett. A57 (1998), 127-137. [15] L. Grover, Quantum Mechanics helps in search for a needle in a haystack, Phys. Rev. Lett., 79 (1997), 325-328. [16] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335 – 388. · Zbl 0631.57005 [17] R. Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics, 41 (1994), no. 12, 2315-2323. · Zbl 0941.81508 [18] Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. · Zbl 0821.57003 [19] A. Yu. Kitaev, Quantum computations: algorithms and error correction, Uspekhi Mat. Nauk 52 (1997), no. 6(318), 53 – 112 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 6, 1191 – 1249. · Zbl 0917.68063 [20] A. Kitaev, Fault-tolerant quantum computation by anyons, quant-ph/9707021. · Zbl 1012.81006 [21] Seth Lloyd, Universal quantum simulators, Science 273 (1996), no. 5278, 1073 – 1078. · Zbl 1226.81059 [22] Gregory Moore and Nicholas Read, Nonabelions in the fractional quantum Hall effect, Nuclear Phys. B 360 (1991), no. 2-3, 362 – 396. [23] C. Nayak, and K. Shtengel, Microscopic models of two-dimensional magnets with fractionalized excitations, Phys. Rev. B, 64:064422 (2001). [24] Chetan Nayak and Frank Wilczek, 2\?-quasihole states realize 2$$^{n}$$$$^{-}$$\textonesuperior -dimensional spinor braiding statistics in paired quantum Hall states, Nuclear Phys. B 479 (1996), no. 3, 529 – 553. · Zbl 0925.81445 [25] J. Preskill, Fault tolerant quantum computation, quant-ph/9712048. [26] N. Read, and E. Rezayi, Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Laudau level, cond-mat/9809384. [27] P. Shor, Algorithms for quantum computation, discrete logarithms and factoring, Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, 124-134. [28] P. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, 2493 (1995). [29] P. Shor, Fault-tolerant quantum computation, Proc. 37th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1996. [30] R. Solvay, private communication. [31] V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. · Zbl 0812.57003 [32] K. Walker, On Witten’s 3-manifold invariants, preprint, 1991 (available at http://www.xmission.com/$$\sim$$kwalker/math/). [33] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351 – 399. · Zbl 0667.57005
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