By Zhenghan Wang

Topological quantum computation is a computational paradigm in accordance with topological levels of subject, that are ruled via topological quantum box theories. during this technique, details is kept within the lowest strength states of many-anyon structures and processed by way of braiding non-abelian anyons. The computational resolution is accessed by way of bringing anyons jointly and looking at the outcome. in addition to its theoretical esthetic charm, the sensible benefit of the topological process lies in its error-minimizing hypothetical undefined: topological levels of subject are fault-avoiding or deaf to so much neighborhood noises, and unitary gates are applied with exponential accuracy. Experimental realizations are pursued in platforms equivalent to fractional quantum corridor beverages and topological insulators. This ebook expands at the author's CBMS lectures on knots and topological quantum computing and is meant as a primer for mathematically susceptible graduate scholars. With an emphasis on introducing uncomplicated notions and present examine, this publication offers the 1st coherent account of the sector, protecting a variety of issues: Temperley-Lieb-Jones conception, the quantum circuit version, ribbon fusion classification concept, topological quantum box thought, anyon concept, additive approximation of the Jones polynomial, anyonic quantum computing types, and mathematical versions of topological stages of subject.

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Over the basis of Q(q±112) given by powers of q112, we may write J(L; q) as a string of integers, which is easily encoded as bit string. Hence with encodings the computation of J(L; q) is a map J(L; q) : {0, 1}* --+ {0, 1}* Can we compute J(L; q) efficiently? The computation can be either quantum me­ chanical or classical. Moreover rather than the exact answer we can ask for an approximation according to various approximation schemes. Therefore there are many ways to formulate the computation of the Jones polynomial as a computing problem.

2) A fusion rule on a label set L is a binary operation ® : L L --+ N L , where N L is the set of all maps from L to N = {0, 1, 2, . . } satisfying the following conditions. First we introduce some notation. Given a, b E L, we will write formally a ® b = E9 N�b c where N�b = (a ® b) (c) . When no confusion arises, we write a ® b simply as ab, so a2 = a ® a. Then the conditions on ® are: for all a, b, c, d E L, (i) (a ® b) ® c = a ® (b ® c) , i. , L xE L N�b N�c = LxE L Nbc N�x (ii) N�1 = N'fa bca (iii) N};b = Nla iSba We say a triple of labels (a, b, c) is admissible if N�b -1- 0.

We have already seen two different normalizations of the Jones representation at r = 4 or level k = 2: the Jones normalization and the Kauffman bracket nor­ malization. There is also a third normalization related to the "(-matrices: " � pA_ (ui) = e •"�'+1"�' = 1 ( 1 + 'Yi +l 'Yi) · J2 The orders of p{(ui) , p}(ui), and PA(ui) are 4, 8, and 16 respectively. Although all three normalizations have the same projective image, their linear images are in general different. _ 1 , the image is E;;� 1 [FRW] .

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