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A Quantum Manual for Computing the Jones Polynomial

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A Quantum Manual for Computing the Jones Polynomial A Quantum Manual for Computing the Jones Polynomial Samuel J. Lomonaco, Jr.a and Louis H. Kauffmanb aDepartment of Computer Science and Electrical Engineering, 1000 Hilltop Circle, University of Maryland Baltimore County (UMBC), Baltimore, MD 21250, USA bDepartment of Mathematics, Statistics, and Computer Science, 851 South Morgan Street, University of Illinois at Chicago, Chicago,IL 60607-7045, USA ABSTRACT The objective of this paper is to give experimentalists a quantum manual for implementing the Aharonov-Jones- Landau algorithm on an architecture of their choice. In particular, we explicitly apply this algorithm to a number of knot examples. 1. INTRODUCTION The objective of this paper is to give experimentalists a quantum manual for implementing the Aharonov-Jones- Landau algorithm on an architecture of their choice. In particular, we explicitly apply this algorithm to a number of knot examples. 2. BEGINNING CALCULATIONS Let b1 and b2 be the generators of the n =3stranded braid group B3. We wish to compute the values of the Jones polynomial J (t) at t = e2πi/k. We begin by defining the following parameters: iπ/(2k) A = e− d =2cos(π/k) sin (πm/k) if 0 <m<k λm = 0 otherwise Then for each of the generators, we construct the following matrices: A 0 (2) (4) B1 =(Φ ρA)(b1)= (A)1 1 = B1 B1 ◦ 0 A 2 2 ⊕ × ⊕ µ ¶ × 1 1 A + A− λ1 A− √λ1λ3 λ2 λ2 (2) (4) B2 =(Φ ρA)(b2)= A 1√λ λ A 1λ (A)1 1 = B2 B2 ◦ − 1 3 A + − 3 ⊕ × ⊕ Ã λ2 λ2 !2 2 × where for matrices X and Y , X Y denotes the direct sum matrix ⊕ X 0 . 0 Y µ ¶ S.J.L.: Email: [email protected], L.H.K.: E-mail: kauff[email protected] 3. THE TREFOIL 3 The trefoil is the closure of the braid β = b1b2.So 3 3 3 (2) (2) (4) (4) (2) (4) (Φ ρA)(β)=B B2 = B B B B = C C ◦ 1 1 2 ⊕ 1 2 ⊕ ·³ ´ ³ ´¸ ·³ ´ ³ ´¸ Let J(t) be the Jones polynomial of the trefoil. Then 2πi/k 2Writhe(β) n 1 J e = A d − Trn [(Φ ρA)(β)] − ◦ ³ ´ 3 1 where n =3,andWrithe(β)=Writhe b1b2 =3+1=4is the sum of the exponents in the braid word 3 β = b b2. 1 ¡ ¢ Thus, 2πi/k 8 2 (2) (4) J e = A d Tr3 C C − ⊕ ³ ´ h i where Tr3 is a special trace defined as follows: (2) (4) 1 (2) (4) Tr3 C C = λ2Tr C + λ4Tr C ⊕ N h i h ³ ´ ³ ´i where "Tr" denotes the standard trace, and where N is a normalizing factor given by N =2λ2 + λ4 Thus, the task of computing J e2πi/k reduces to that of computing the standard traces of the matrices C(2) and C(4). If one computes J e2πi/k for k =2, 3, 4, 5, then one can use polynomial interpolation to compute ¡ ¢ the coefficients of the Jones polynomial. Since the coefficients of the Jones polynomial are integers, one simply rounds the interpolated values¡ to the¢ nearest interger. If one is close enough to the actual values of J e2πi/k , then these rounded numbers will be the correct coefficients. ¡ ¢ The reader can now verify that the Jones polynomial for the trefoil is: J(t)=t 1+t2 t3 − ¡ ¢ 4. OTHER KNOTS This same procedure can be used for other knots that are the closure of 3-straned braids. One simply takes abraidwordβ whose trace is the knot, and then uses the formulas of the previous section to determine the matrices C(2) and C(4). The reader can verify calculations with the table given below: Knot J(t) 3-strand braid word 2 3 3 t 1+t t β = b b2 − 1 ¡ ¢ 2 2 3 4 1 1 t− 1 t + t t + t β = b2b− b2b− − − 1 1 ¡ ¢ 2 2 3 4 5 5 t 1+t t + t t β = b b2 − − 1 ¡ ¢ 2 3 4 5 2 2 1 t 1 t +2t t + t t β = b b b1b− − − − 1 2 2 ¡ ¢ REFERENCES 1. Aharonov, D., V. Jones, Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, quant-ph/0511096. 2. Birman, Joan S., "Braids, Links, and Mapping Class Groups," Princeton University Press, (1975). 3. Crowell, Richard H., and Ralph H. Fox, "Inroduction to Knot Theory," Springer-Verlag, (1977). 4. Famy, Amr, Steffen Glasser, Louis Kauffman, Samuel Lomonaco, Raimund Marx, John Myers, and Andreas Spoerl, rough draft, (2006). 5. Garnerone, Silvano, Annalisa Marzuoli, and Mario Rasetti, Quantum automata, braid group and link polynomials, http://arxiv.org/abs/quant-ph/0601169 6. Chen, Goong, Louis H. Kauffman, and Samuel J. Lomonaco, (editors), "The Mathematics of Quantum Computation and Quantum Technology," Chapman Hall, (to appear in 2007). 7. Jones, V.F.R.J, A polynomial invariant for links via von Neumann algebras, Bull.Amer.Math.Soc. 129 (1985) 103-112. 8. Jones, V.F.R., A new knot polynomial and von Neumann algebras, Notices of AMS 33 (1986) 219-225. 9. Jones, V.F.R., Index for subfactors, Invent. Math 72 (1983), 1—25. 10. Jones, V.F.R., Braid groups, Hecke Algebras and type II factors,in"Geometric methods in Operator Algebras," Pitman Research Notes in Math., 123 (1986), 242—273. 11. Kauffman, Louis H., "Knots and Physics," (third edition), World Scientific, (2001). 12. Kauffman, Louis H., and Sostenes L. Lins, "Temperley-Lieb Recoupling and Invariants of 3- Manifolds," Princeton University Press, (1994) 13. Kauffman, Louis, Quantum Computing and the Jones polynomial,in"Quantum Computation and Information," edited by Samuel J. Lomonaco, Jr. and Howard E. Brandt, AMS CONM/305, (2002), pp.101 - 137. (math.QA/0105255). 14. Kauffman, Louis H., State Models and the Jones Polynomial, Topology 26, (1987) 395-407. 15. Kauffman, Louis H., "On Knots," Annals of Mathematics Studies Number 115, Princeton University Press (1987). 16. Kauffman, Louis H., New invariants in the theory of knots, Amer. Math., Monthly Vol.95,No.3,March 1988. pp 195-242. 17. Kauffman, Louis H., Statistical mechanics and the Jones polynomial, AMS Contemp. Math. Series (1989), Vol. 78. pp. 263-297. 18. Kauffman, Louis H., and Samuel J. Lomonaco, Jr., q-Deformedspinnetworks,knotpolynomials, and anyonic topological quantum computation, Journal of Knot Theory and Its Ramifications, (to appear). (http://arxiv.org/abs/quant-ph/0606114 ) 19. Kauffman, Louis H., and Samuel J. Lomonaco, Jr., Spin networks and anyionic topological comput- ing, Proc. SPIE, Vol. 6244, (2006). (http://xxx.lanl.gov/abs/quant-ph/0603131 ) 20. Kauffman, Louis H., and Samuel J. Lomonaco, Jr., Quantum knots, Proc. SPIE, (2004). (http://xxx.lanl.gov/abs/quant-ph/0403228 ) 21. Lomonaco, Samuel J., Jr., and Louis H. Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE, Vol. 6244, (2006). (http://xxx.lanl.gov/abs/quant-ph/0605004 ) 22. Lomonaco, Samuel J., Jr., and Louis H. Kauffman, Quantum algorithms for the Jones polynomial, (to appear) 23. Lomonaco, Samuel J., Jr., "Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium," AMS PSAPM 58, (2000). 24. Lomonaco, Samuel J., Jr., (ed.), "Low Dimensional Topology," AMS CONM/20, (1981). 25. Murasugi, Kunio, "Knot Theory and Its Applications," Birkhauser, (1996)..
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