louis h. kauffman and samuel j. lomonaco jr- spin networks and anyonic topological computing

8/3/2019 Louis H. Kauffman and Samuel J. Lomonaco Jr- Spin Networks and Anyonic Topological Computing

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Spin Networks and Anyonic Topological Computing

Louis H. Kauffmana and Samuel J. Lomonaco Jr.b

a Department of Mathematics, Statistics and Computer Science (m/c 249), 851 South MorganStreet, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA

b Department of Computer Science and Electrical Engineering, University of Maryland

Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA

quant-ph/0603131 and quant-ph/0606114

www.math.uic.edu/~kauffman/Unitary.pdf

Spin Networks and Anyonic

Topological Quantum Computing

L. H. Kauffman, UIC


2/79

Quantum Mechanics in a Nutshell

1. (measurement free) Physical processesare modeled by unitary transformationsapplied to the state vector: |S> -----> U|S>

0. A state of a physical systemcorresponds to a unit vector |S> in a

complex vector space.

2. If |S> = z1|e1> + z2|e2> + ... + zn|en>

in a measurement basis {|e1>,|e2>,...,|en>}, thenmeasurement of |S> yields |ei> with

probability |zi|^2.

U


3/79

U

U*

U* UPsi*Psi =

Psi =

U

Preparation,Transformation,

Measurement.


4/79

Qubit

a|0> + b|1>

|0>|1>

measure

prob = |a|^2 prob = |b|^2

A qubit is the quantum version ofa classical bit of information.


5/79

|0> |1>|0>

|0> |1>|0>

|0>

|1>

-|1>

|0> |1>

|0>

|0>|1>

-|1>

Mach-Zender Interferometer

H = [ ]1 11 -1 /Sqrt(2) M = [ ]11

0

0

HMH = [ ]1 00 -1


6/79

Quantum Gates

are unitary transformationsenlisted for the purpose of computation.

1 0 0 0

0 0 0

0 0 0

0 0 0

1

1

1CNOT =

CNOT|00> = |00>CNOT|01> = |01>CNOT|10> = |11>CNOT|11> = |10>


7/79

1. A good example of a quantum algorithm.

2. Useful for the quantum computation ofknot polynomials such as the Jones polynomial.

Quantum Computation of the Traceof a Unitary Matrix

U


8/79

U

H|0>

|phi>

Measure

Hadamard Test

|0>

|0> occurs with probability1/2 + Re[]/2

H


9/79

Grovers Algorithm (1996)[O(Sqrt(N)) time, O(log N) storage space]

Given an unsorted database with N entries.

{0,1,2,...,N-1}

Form N-dimensional state space V.H an observable acting on V

with N distinct eigenvalues.{|0>,|1>,|2>,...|N-1>} basis for V.

Problem: Find a particular element w in

the database.


12/79

a

rXiv:quant-ph

/9508027v2

25Jan1996

Polynomial-Time Algorithms for Prime Factorization

and Discrete Logarithms on a Quantum Computer

Peter W. Shor

Abstract

A digital computer is generally believed to be an efficient universal computing

device; that is, it is believed able to simulate any physical computing device with

an increase in computation time by at most a polynomial factor. This may not betrue when quantum mechanics is taken into consideration. This paper considers

factoring integers and finding discrete logarithms, two problems which are generally

thought to be hard on a classical computer and which have been used as the basis

of several proposed cryptosystems. Efficient randomized algorithms are given for

these two problems on a hypothetical quantum computer. These algorithms take

a number of steps polynomial in the input size, e.g., the number of digits of the

integer to be factored.

Keywords: algorithmic number theory, prime factorization, discrete logarithms,

Churchs thesis, quantum computers, foundations of quantum mechanics, spin systems,Fourier transforms

AMS subject classifications: 81P10, 11Y05, 68Q10, 03D10


13/79

Demonstrations


14/79

Universal Gates

G : V V

A two-qubit gate G is a unitary linear mapping

where V is

a two complex dimensional vector space. We say that the gate G is universal for quantum computation(or just universal) ifG together with local unitarytransformations (unitary transformations from V to V) generates all unitarytransformations of the complex vector space of dimension 2n to itself. It iswell-known [44] that CNOT is a universal gate.

V V

Explicit gate realization in the basis fj0i; j1ig:

H D 1p2

1 1

1 1

; S D

1 0

0 i

; T D

1 0

0 ei=4

Local Unitaries are generated (up to density) bya small number of gates.


15/79

A gate G, as above, is said to be entangling if there is a vector

| = | | V V

such that G| is not decomposable as a tensor product of two qubits.Under these circumstances, one says that G| is entangled.

In [6], the Brylinskis give a general criterion ofG to be universal. They provethat a two-qubit gate G is universal if and only if it is entangling.

A gate G is universaliff

G is entangling.


16/79

An Entangled State


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Remark. A two-qubit pure state

| = a|00 + b|01 + c|10 + d|11

is entangled exactly when (ad bc) = 0. It is easy to use this fact to checkwhen a specific matrix is, or is not, entangling.

An Entanglement Criterion

R|00 = (1/

2)|00 (1/

2)|11,R|01 = (1/

2)|01 + (1/

2)|10,

|10 = (1/

2)|01 + (1/

2)|10,R|11 = (1/

2)|00 + (1/

2)|11.

The Bell States


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Braiding and the Yang-Baxter Equation

=

RIR I

RI

RI

RI

R I

R I

R I

(R I)(IR)(R I) = (IR)(R I)(IR).


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R: V

V

V

V

Let V be atwo complex dimensional vector space.

Universal gates can be constructedfrom certain solutions to theYang-Baxter Equation

(R I)(IR)(R I) = (IR)(R I)(IR).

Braiding Operators are Universal

Quantum Gates


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R =

1/

2 0 0 1/

2

0 1/

2 1/

2 0

0 1/

2 1/

2 0

1/

2 0 0 1/

2

R =

a 0 0 00 0 b 00 c 0 00 0 0 d

R0 =

1 0 0 00 0 1 0

0 1 0 00 0 0 1

R =

0 0 0 a0 b 0 00 0 c 0d 0 0 0

Representative Examples ofUnitary Solutions to the

Yang-Baxter Equation that are Universal Gates.

Bell Basis Change Matrix

Swap Gatewith Phase


21/79

Issues

1. Giving a Universal Gate that istopological gives PARTIAL

topological quantum computingbecause the U(2) local operationshave not been made topological.

2. Nevertheless, Yang-Baxtergates are interesting to

construct and

help to discussTopological Entanglementversus

Quantum Entanglement.


22/79

Quantum Entanglement andTopological Entanglement

An example of Aravind [1] makes the possibility of such a connection even more tantalizing.Aravind compares the Borromean rings (see figure 2) and the GHZ state

| = (|1|2|3 |1|2|3)/

2.

Is the Aravind analogy onlysuperficial?!

(|000> - |111>)/Sqrt(2)


23/79

| = (1/2)(|000 + |001 + |101 + |110)

Consider this state.

Observation in any coordinateyields entangled and unentangled

states with equal probability.

| = (1/2)(|0(|00 + |01) + |1(|01 + |10)

e.g.

First coordinate measurement

gives|00> + |01> and

|01> + |10>with equal probability.


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Do we need Quantum Knots?

rr$$$$$$

$X

~~~~~~ x"!#$$$

+

c

$$$$$$

$

333

ll

|

Observing a Quantum Knot

a|K> + b|K>

K: probability |a|^2

K:probability |b|^2

K K


25/79

-

I

II

III

-1 2

3 1-1

=

=

=

s

s s

s

Braid Generators

1s1-1

s = 1

1s 2s 1

s 2s 1s 2s=

1s 3

s 1s

3s=

Hopf Link

Figure Eight Knot

Trefoil Knot

Knots and Links


26/79

[


27/79

-

|

-[

Identity

| >

< |

< | >

< || > =

U

=

=

< || > 1

< || >1

=

=

P

Q

{ }

}{

=

PQP P=

| >

| >

| > | >

| > | >=

| >

< |

| |

| | -

| | |

| |The Key to Teleportation

The Temperley-Lieb Category

QPQ=Q


28/79

P = > < = >< = P

QQ = { } Q

PQP = >< } { > < = < } { > > < = < } { > PQPQ = { > < } Q

Any two one-dimensionalprojectors generate a

Temperley-Lieb algebra.

This trick can be used to manufactureunitary representations of the three-strand

braid group.


29/79

Diagrammatic Matrices, Knots andTeleportation

a b

a b

M

M

a b

a b

N = M Maiib

a

b

a

i

b

Na

b

a

b

Figure 5 - Matrix Composition


30/79


31/79


32/79


33/79

|Cup> = ! M |i>|b>

#

|Cup>


34/79

The Topology of Teleportation

| = |, H H.

|00> + |11> 1 0

0 1

|!>

"|!>


35/79

Theorem. If g = a + bu and h = c + dv are pure unit quaternions,then,without loss of generality, the braid relation ghg = hgh is true if and only ifh = a + bv, and g(v) = h1(u). Furthermore, given that g = a + bu and

h = a+ bv, the condition g(v) = h1(u) is satisfied if and only ifu v =a2b2

2b2

when u = v. If u = v then then g = h and the braid relation is triviallysatisfied.

SU(2) Representations of the Artin BraidGroup

g = a + buh= a + bv

u v = (a^2 - b^2)/2b^2


36/79

An Example. Letg = ei = a + bi

where a = cos() and b = sin(). Let

h = a + b[(c2 s2)i + 2csk]

where c2 +s2 = 1 and c2s2 = a2b2

2b2. Then we can reexpress g and h in matrix

form as the matrices G and H. Instead of writing the explicit form of H, we

write H= FGF

where F is an element ofSU(2) as shown below.

G =

ei 00 ei

F =

ic is

is ic


37/79

g = e7i/10

f= i + k

h = frf1

SU(2) Fibonacci Model

2 + = 1.

{g,h} represents 3-strand braids,generating a dense subset of SU(2).

= fgf-1


38/79

We shall see that the representationlabeled SU(2) Fibonacci Model

in the last slideextends beyond SU(2) to

representations of many-strandedbraid groups rich enough

to generate quantum computation.


39/79

Quantum Hall Effect


40/79

Fractional Quantum Hall Effect(Cambridge Univ Website)


41/79

The quasi-particle theory is connected

with Chern-Simons Theory and itexplains the FQHE on the basis of

anyons: particles that have non-trivial(not +1 or -1) phase change when they

exchange places in the plane.


42/79

Braiding Anyons

Recoupling

Process Spaces


43/79

F R

B = F RF-1

F-1

Non-Local Braiding is Induced

via Recoupling


44/79

Process Spaces Can be Abitrarily Large.

With a coherent recoupling theory, alltransformations are in the

representation of one braid group.


45/79

Mathematical Models for RecouplingTheory with Braiding come from a

Combination ofPenrose Spin Networks and

Knot Theory.

See Temperley Lieb Recoupling Theoryand Invariants of Three-Manifolds by

L. Kauffman and S. Lins, PUP, 1994.


46/79


47/79


48/79

-

I

II

III

-1 2

3 1-1

=

=

=

s

s s

s

Braid Generators

1s

1

-1s = 1

1s 2s 1

s 2s 1s 2s=

1s 3

s 1s

3s=

Hopf Link

Figure Eight Knot

Trefoil Knot

Knots and Links


49/79

AA

-1A

-1A

A

-1A

< > = A < > + < >-1

A

< > = A< > + < >-1

A

< K >=

S

< K|S > ||S||1.

Bracket Polynomial Model forJones Polynomial


50/79

K U U'

-

-

A1 < K > A < U >= (A2 A2) < U >

< U >= A3

< U >= (A3)2 = A6

< K >=

A5

A3 + A

7.

Trefoil Calculation

-


51/79

- -[

-

...

...

n strands

=n

n= (A )-3

t( ) ~(1/{n}!)

Sn

~

=

A A-1

=-A

2 -2- A

= +

{n}! = Sn

(A )t( )-4

=n

n

= 0

=d

45

= 1/

= /n n+1

n 1 1 n 1 1

n

1

=

2

1 =-1 = 0 0

n+1 = n

-n-1

a b

c

ij

k

a b

c

i + j = a

j + k = b

i + k = c

q-Deformed Spin Networks


52/79

( )3

1

1

2..UUU

RBorrom= -( )

3

1UU

Trefoil= ( )

2

1

1

2UUU

EightFigure= -

-

Untying Knots by NMR: first experimental implementationof a quantum algorithm for approximating the Jones polynomial

Raimund Marx1, Andreas Sprl , Amr F. Fahmy , John M. Myers , Louis H. Kauffman ,

Samuel J. Lomonaco, Jr. , Thomas-Schulte-Herbrggen , and Steffen J. Glaser

1 2 3 4

5 1 1

1

2

Department of Chemistry, Technical University Munich,Lichtenbergstr. 4, 85747 Garching, Germany

Harvard Medical School, 25 Shattuck Street, Boston, MA 02115, U.S.A.

3

4

5

Gordon McKay Laboratory, Harvard University, 29 Oxford Street, Cambridge, MA02138, U.S.A.

University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, U.S.A.

University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, U.S.A.

trace-closed

braid

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. Sprl, A. F. Fahmy, J. M. Myers, L. H. Kauffman, S. J. Lomonaco, Jr., T. Schulte-Herbrggen, and S. J. Glaser: paper in preparation3) Vaughan F. R. Jones, Bull. Amer. Math. Soc., (1985), no. 1, 103-1114) 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, 58 7-622

knotor

link

unitary

matrix

controlledunitarymatrix

NMRpulse

sequence

NMRexperiment

Jonespolynomial

roadmap of thequantum algorithm

example #1Trefoil

example #2Figure-Eight

example #3Borromean rings

s1

s1

s1

s1s2

-1

s1

s2-1

s1s2

-1

s1s2

-1

s1s2

-1

+- - qq

q

q ii ee)4sin(

)2sin(- q

q

qqie)4sin(

)2sin()6sin(

- q

q

qqie

)4sin(

)2sin()6sin(+- - qq

q

q ii ee)4sin(

)6sin(

=U2

- qie

+- - qq

q

q ii ee)2sin(

)4sin(0

0

=U1

Step #1:from the 2x2 matrixto the 4x4 matrix :

UcU

cU= ( )U001

Step #2:application of on the

NMR product operator :cU

I1x

cU = ( )U0 01cU I1x 21 ( )100

1 ( )U001

=21 ( )0

0 U

U

Step #3:

measurementof and :I I1x 1y

I1x ( )00 U

U

{ }tr = 21 tr U( { } )I1y ( )0

0 U

U

{ }tr = 21 tr U( { } )21

21

IS

cU2-1

cU1 cU2-1

cU1 cU2-1

cU1I

ScU2

-1cU1 cU2

-1cU1

IS

cU1 cU1 cU1

I

S

z

-b

z

-a

apJ

IS

cU1

means

I

S

z

b

y

g p+

y

-g

apJ

y

-p

z

aIS

cU2-1

means

I

S

z

-b

y

g

z

-a

y

-g

apJ

IS

cU2

means

I

S

z

b

y

p

apJ

y

-p

z

aIS

cU1-1

means

Jones PolynomialTrefoil":

Jones PolynomialFigure-Eight":

Jones PolynomialBorromean rings":

- + - A 3A 2A2 13

+ -A A8 4

+ -A A-4-8

+ 4A0

- + - A 3A 2A-1-2-3

+A0

A is defined as a closed, non-self-intersecting curve

that is embedded in three dimensions.

knot

example: construction of the Trefoil knot:

make aknot

fuse thefree ends

make itlook nice

st art with a r ope end up with a Trefoil

J. W.Alexander proved, that any knot can be representedas a closed braid (polynomial time algorithm)

1 s1 s2s1-1 s2

-1

generators of the 3 strand braid group:

radieq-

=

It is well known in knot theory, how to obtain the unitary matrix representationof all generators of a given braid goup (see Temperley-Lieb algebra and pathmodel representation). The unitary matrices U and U , corresponding to the

generators and of the 3 strand braid group are shown on the left, where the

variable is related to the variable of the Jones polynomial by: .

The unitary matrix representations of and are given by U and U .

The knot or link that was expressed as a product of braid group generators cantherefore also be expressed as a product of the corresponding unitary matrices.

1 2

1 2s s

q

s s

A A-

1- - -

1 1 1

1 2 1 2

Instead of applying the unitary matrix we apply its controlled variant .This matrix is especially suited for NMR quantum computers [4] and otherthermal state expectation value quantum computers: you only have to apply

to the NMR product operator and measure and in order to obtain

the trace of the original matrix .

U, cU

cU I I I

U1x 1x 1y

.

Independent of the dimension of matrix you only need ONE extra qubit for the

implementation of as compared to the implementation of itself.U

cU U

The measurement of I I1x 1y

and can be accomplished in one single-scan experiment.

All knots and links can be expressed as a product of braid group generators (seeabove). Hence the corresponding NMR pulse sequence can also be expressed asa sequence of NMR pulse sequence blocks, where each block corresponds to thecontrolled unitary matrix of one braid group generator.cU .

This modular approach allows for an easy optimization of the NMR pulsesequences: only a small and limited number of pulse sequence blocks have to

be optimized. .

Comparison of experimental results, theoretical predictions, and simulated ex-periments, where realisitic inperfections like relaxation, B field inhomogeneity,

and finite length of the pulses are included.1

.

The Jones Polynomials can be reconstructed out of the NMR experiments by:

For each data point, four single-scan NMR experiments have been performed:measurement ofI I I I

1x 1y 1x 1y, measurement of , reference for , and reference for .

If necessary each data point can also be obtained in one single-scan experimentby measuring amplitude and phase in a referenced setting. .

V (A)=( A ) ( { } A [( A A ) 2])- +tr U - - -3 ( ) ( ) 2 2 2- -w L I L

L

where: ( ) is the writhe of the knot or link{ } is determined by the NMR experiments

( ) is the sum of exponents in the braid wordcorresponding to the knot or link

w L Ltr UI L

L

A A A+ --4 -12 -16( )

- -A A2 -2( )

-


53/79

- -[

-

...

...

n strands

=n

n= (A )-3

t( ) ~(1/{n}!)

Sn

~

=

A A-1

=-A

2 -2- A

= +

{n}! = Sn

(A )t( )-4

=n

n

= 0

=d

45

= 1/

= /n n+1

n 1 1 n 1 1

n

1

=

2

1 =-1 = 0 0

n+1 = n

-n-1

a b

c

ij

k

a b

c

i + j = a

j + k = b

i + k = c

q-Deformed Spin Networks


54/79

Projectors are Sums overPermuations, Lifted to

Braids and Expanded viathe Bracket into the

Temperley Lieb Algebra


55/79

-

-

[ [

[

Trinion

Topological Quantum Field

Theory

Process Spaces on SurfacesLead to Three-Manifold

Invariants.

-

--


56/79

a b

c

d

e fa b

c

V( )

V( )

=

-

--

Process Vector Spaces andRecoupling


57/79

A B

C

A B

C

=


58/79

= R

-

-[

=

-

-[

F

-

FF

F

FF=

R

R

R

F

F

F

Braiding, Naturality, Recoupling,Pentagon and Hexagon --

Automatic Consequences of theConstuction


59/79

F R

B = F RF-1

F-1

Non-Local Braiding is Induced

via Recoupling

[


60/79

*

P P P P

P

* *

*

*

| 0 > | 1 >

1110

11110

dim(V ) = 2

dim(V ) = 1

P P P

P

P

P

P P P PP P

P

P

=

Forbidden

-

-

-

= 1/

-

--

- -

- --

-

FibonacciModel

Temperley Lieb

Representation ofFibonacci Model

A = e3i/5.

P


61/79

ICONICS

In the Fibonacci Model we have

one particle P that interactsitself to produce either P or * (nothing).This is analogous to the logical particle

representing an act of distinction

(or registration, measurement, ...)

of G. Spencer-Brownthat interacts with itself in two ways:


62/79

,: , , ...

Digression:The Re-entering Mark as

Iconic for Recursive Traceand Lambda CalculusFixed Point.

AAA =

=

=Hence


63/79


64/79

Fibonacci Model


65/79

A = e3

i/5

.

= = 1 +

5 2

F=

1/ 1/1/ 1/

=

R =

A4 00 A8

=

e4i/5 00 e2i/5

.

= A2 A2

The Simple, yet Quantum Universal,Structure of the Fibonacci Model

S i N k G i


66/79

Spin Network Gymnastics


67/79


68/79

=

= ==

= 1/ = ( 1/)

( 1/) ( 1/)

= 12

= 1/

= ( 1/) 2 /


69/79

a b

c d+

+=

=

a= a=

1/

b= /

2= b

b = /

= c c =2

= d =d /2

/


70/79

ac d

a

=a

= ( , , )a c d

a a= = a

-

= ( , , )

a

b

a c

a

c dd

a

a

b

-

{ }a bc d ij=j

aa

bb

cc d

d

ij

b

c d

a

=k [ ]Tet a bc d iki

Closure, Bubble andRecoupling


71/79

{ }a bc d ij=j

a ab b

cc dd

i jk

= j

( , , )a

( , , )c d

b j j j

j

k

k

{ }a b

c d

i

j

= ( , , )a ( , , )c d

b{ }a bc d ik k kk

=( , , )

{ }a bc d ik [ ]Tet

abc d ik

( , , )k kdca b

j

j

k

The 6-j Coefficients


72/79

a bc

a ab b

c c

(a+b-c)/2 (a'+b'-c')/2

x' = x(x+2)

a bc

=

= (-1) A

Local Braiding


73/79

a b

c

a b

c

=

( , , )

a b c

ca b

Redefining the Vertex is the key to obtaining

Unitary Recoupling Transformations.

= ( , , )

a

ba c

a

c

a

b

a

a

b c

a

= a b c

a

b c

a

=

a

( , , )a cb

a

b c

a

b c

a


74/79

=ja

ab

b

cc dd

i

k

= j j

k

k

=

d

= [ ]ModTeta bc d

ia b

c di j

=b

c d

ai

j

a b c d

a b

c di

a b

c di

j

a b c

j j

k

k k

j

j

da b

c di

a b c

j j

j

j

da b c

j j

New Recoupling Formula


75/79

a b

c di j=

j

aa

b

cc d

d

ij

a b

c di j

=b

c d

ai

j

a b c d

M[a,b,c,d]i j

=a b

c di j a b

c d

=b

c

d

a

ij

a b c d

b

c d

ai

j

a b c d

a b

c d

T -1

==

- -

The Recoupling Matrix isReal Unitary at Roots of

Unity.

,


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A = ei/2r

Theorem. Unitary Representations of theBraid Group come from Temperley Lieb

Recoupling Theory at roots of unity.

Sufficient to Produce Enough Unitary

Transformations for QuantumComputing.

Quantum Computation of


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Quantum Computation ofColored Jones Polynomials and

WRT invariants.

B

P(B)

=

0 0

0

=x

y,

xy 0

B(x,y)

0 0

0

=

a a

a a a a

a a

a a

=x

y,

xy 0

B(x,y)

a a

=0

a a

B(0,0)0 0

= 0a

b

if b = 0

= B(0,0) a( ) 2

Need to compute a diagonalelement of a unitary transformation.

Use the Hadamard Test.


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A4 -4

= A + +

A4-4= A+ +

- =4

A A-4

-( ) -( )

- = 4A A -4-( ) -( )

= A 8

Colored Jones Polynomial for n = 2 isSpecialization of the

Dubrovnik version ofKauffman polynomial.


79/79

Will these models actually be used

for quantum computation?Will quantum computation actually happen?

Will topology play a key role?Time will tell.

louis h. kauffman and samuel j. lomonaco jr- spin networks and anyonic topological computing

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