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Topological Quantum Information and the Jones Polynomial

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Topological Quantum Information and the Jones Polynomial TOPOLOGICAL QUANTUM INFORMATION AND THE JONES POLYNOMIAL Louis H. Kauffman Math, UIC 851 South Morgan St. Chicago, IL 60607-7045 <[email protected]> Arxiv: 1001.0354 Background Ideas and References Topological Quantum Information Theory (with Sam Lomonaco) quant-ph/0603131 and quant-ph/0606114 Spin Networks and Anyonic TQC arXiv:0805.0339 Quantum Knots and Mosaics arXiv:0910.5891 arXiv:0804.4304 The Fibonacci Model and the Temperley-Lieb Algebra arXiv:0706.0020 A 3-Stranded Quantum Algorithm for the Jones Polynomial arXiv:0909.1080 NMR Quantum Calculations of the Jones Polynomial Authors: Raimund Marx, Amr Fahmy, Louis Kauffman, Samuel Lomonaco, Andreas Spörl, Nikolas Pomplun, John Myers, Steffen J. Glaser arXiv: 0909.1672 Anyonic topological quantum computation and the virtual braid group. H. Dye and LK. Untying Knots by NMR: first experimental implementation of a quantum algorithm for approximating the Jones polynomial Raimund Marx1, Andreas Spörl1, Amr F. Fahmy2, John M. Myers3, Louis H. Kauffman4, Samuel J. Lomonaco, Jr.5, Thomas-Schulte-Herbrüggen1, and Steffen J. Glaser1 3 1Department of Chemistry, Technical University Munich, Gordon McKay Laboratory, Harvard University, 29 Oxford Street, Cambridge, MA 02138, U.S.A. 4 Lichtenbergstr. 4, 85747 Garching, Germany University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, U.S.A. 5 2Harvard Medical School, 25 Shattuck Street, Boston, MA 02115, U.S.A. University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, U.S.A. A knot is defined as a closed, non-self-intersecting curve roadmap of the example #1 example #2 example #3 that is embedded in three dimensions. quantum algorithm Trefoil Figure-Eight Borromean rings example: “construction” of the Trefoil knot: knot or link make a fuse the make it “knot” free ends “look nice” start with a rope end up with a Trefoil J. W. Alexander proved, that any knot can be represented s1 s - as a closed braid (polynomial time algorithm) s1 1 s 1 “trace- -1 2 generators of the 3 strand braid group: s2 s1 s -1 -1 closed” 1 -1 1 s s s s s s2 1 1 2 2 braid 1 s1 -1 s1 s2 -1 s2 3 2 3 It is well known in knot theory, how to obtain the unitary matrix representation -1 -1 UTrefoil = (U1 ) U Figure-Eight = (U 2 ×U1 ) U Borrom.R. = (U 2 ×U1 ) of all generators of a given braid goup (see “Temperley-Lieb algebra” and “path model representation”). The unitary matrices U1 and U2, corresponding to the æ ö æ iq sin(6q ) -iq ö -iq iq sin(6q )sin(2q ) generators s1 and s2 of the 3 strand braid group are shown on the left, where the unitary ç e 0 ÷ ç - e + e - e ÷ -iq ç ÷ ç sin(4q ) sin(4q ) ÷ variable “q” is related to the variable “A” of the Jones polynomial by: A = e rad. matrix U = U = The unitary matrix representations of s-1 and s-1 are given by U-1 and U-1. 1 ç ÷ 2 ç ÷ 1 2 1 2 iq sin(4q ) -iq sin(6q )sin(2q ) iq sin(2q ) -iq ç 0 - e + e ÷ ç - eiq - e + e ÷ ç sin(2q ) ÷ ç sin(4q ) sin(4q ) ÷ The knot or link that was expressed as a product of braid group generators can è ø è ø therefore also be expressed as a product of the corresponding unitary matrices. Step #1: Step #2: Step #3: Instead of applying the unitary matrix U, we apply it’s controlled variant cU. from the 2x2 matrix U application of cU on the measurement This matrix is especially suited for NMR quantum computers [4] and other thermal state expectation value quantum computers: you only have to apply to the 4x4 matrix cU: NMR product operator I1x : of I 1x and I 1y : cU to the NMR product operator I1x and measure I1x and I1y in order to obtain controlled † † 0 U the trace of the original matrix U. † 1 0 1 0 1 1 0 tr I 1 = 1 Â (tr{U}) unitary 1 0 cU I 1x cU = 2 1x 2 (U 0 ) 2 cU = ( 0 U ) ( 1 0 ) ( 0 U ) { } Independent of the dimension of matrix U you only need ONE extra qubit for the matrix ( 0 U ) † 1 0 U 0 U † implementation of cU as compared to the implementation of U itself. = tr I 1 = 1 Á(tr{U}) 2 (U 0 ) { 1y 2 (U 0 )} 2 The measurement of I1x and I1y can be accomplished in one single-scan experiment. -b b -b b All knots and links can be expressed as a product of braid group generators (see I I I I above). Hence the corresponding NMR pulse sequence can also be expressed as z z z z a sequence of NMR pulse sequence blocks, where each block corresponds to the I -a I -1 p -p a I g -a -g I -1 g+p -p a -g NMR cU cU cU cU . S 1 S 1 S 2 S 2 controlled unitary matrix cU of one braid group generator. S S S S pulse means z means y y z means y z y means y y z y a a a a sequence pJ pJ pJ pJ This modular approach allows for an easy optimization of the NMR pulse I cU cU cU I cU cU -1 cU cU -1 I cU cU -1 cU cU -1 cU cU -1 sequences: only a small and limited number of pulse sequence blocks have to S 1 1 1 S 1 2 1 2 S 1 2 1 2 1 2 be optimized. Comparison of experimental results, theoretical predictions, and simulated ex- periments, where realisitic inperfections like relaxation, B1 field inhomogeneity, and finite length of the pulses are included. NMR experiment For each data point, four single-scan NMR experiments have been performed: measurement of I1x , measurement of I1y , reference for I1x , and reference for I1y . If necessary each data point can also be obtained in one single-scan experiment by measuring amplitude and phase in a referenced setting. Jones Polynomial Jones Polynomial Jones Polynomial The Jones Polynomials can be reconstructed out of the NMR experiments by: “Trefoil": “Figure-Eight": “Borromean rings": 3 -w(L) I(L) 2 -2 2 V (A)=(-A ) (tr{U}+A [(-A -A ) -2]) Jones L ( -4 + -12 - -16) + 8 - 4 - 3 + 2 - 1 polynomial A A A A A A 3A 2A 2 -2 -8 -4 -3 -2 -1 where: w(L) is the writhe of the knot or link L (- A - A ) + A - A - A + 3A - 2A tr{U} is determined by the NMR experiments 0 0 I(L) is the sum of exponents in the braid word + A + 4A corresponding to the knot or link L References: 1) 1) L. Kauffman, AMS Contemp. Math. Series, 305, edited by S. J. Lomonaco, (2002), 101-137 (math.QA/0105255) 2) R. Marx, A. Spörl, A. F. Fahmy, J. M. Myers, L. H. Kauffman, S. J. Lomonaco, Jr., T. Schulte-Herbrüggen, and S. J. Glaser: paper in preparation 3) Vaughan F. R. Jones, Bull. Amer. Math. Soc., (1985), no. 1, 103-111 4) J. M. Myers, A. F. Fahmy, S. J. Glaser, R. Marx, Phys. Rev. A, (2001), 63, 032302 (quant-ph/0007043) 5) D. Aharonov, V. Jones, Z. Landau, Proceedings of the STOC 2006, (2006), 427-436 (quant-ph/0511096) 6) M. H. Freedman, A. Kitaev, Z. Wang, Commun. Math. Phys., (2002), 227, 587-622 Quantum Inf Process S. J. Lomonaco, L. H. Kauffman Keywords Quantum knots Knots Knot theory Quantum computation DOI 10.1007/s11128-008-0076-7Quantum algorithms Quantum· vortices· · · · Mathematics Subject Classification (2000) Primary 81P68 57M25 81P15 57M27 Secondary 20C35 · · · · 1 Introduction The objective of this paper is to set the foundation for a research program on quantum knots.1 For simplicity of exposition, we will throughout this paper frequently use the term “knot” to mean either a knot or a link.2 In part 1 of this paper, we create a formal system (K, A) consisting of (1) A graded set K of symbol strings, called knot mosaics, and (2) A graded subgroup A, called the knot mosaic ambient group, of the group of all Quantum knotspermutations of the set of knotandmosaics K. mosaics We conjecture that the formal system (K, A) fully captures the entire structure of tame knot theory. Three examples of knot mosaics are given below: with SamuelSam J. Lomonaco Louis H. Kauffman Lomonaco · Each of these knot mosaics is a string made up of the following 11 symbols called mosaic tiles. An example of an element in the mosaic ambient group A is the mosaic Reidemeister 1 move illustrated below: Each mosaic is a tensor product of elementary tiles. 1 A PowerPoint presentation of this paper can be found at http://www.csee.umbc.edu/~lomonaco/Lectures. html. 2 For references on knot theory, see for example [4,10,13,20]. 123 © Springer Science+Business Media, LLC 2008 Abstract In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodi- ment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope.
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