22 1. TEMPERLEY-LIEB-JONES THEORIES In this basis, ρA(σ1) = ρ(σ3) = Å e−4πi/5 0 0 e3πi/5 ã ρA(σ2) = Å φ−1e4πi/5 φ−1/2e−3πi/5 φ−1/2e−3πi/5 −φ−1 ã It is unknown whether there exists a braid σ such that ρA(σ) = λ ( 0 1 1 0 ) for some scalar λ, i.e., whether the NOT gate in quantum computing can be realized exactly in the Fibonacci theory up to an overall phase. 1.6. Yamada and chromatic polynomials There is a close relationship between graphs and alternating links through the medial graph construction. The Jones polynomial of an alternating link L is the same as the Tutte polynomial T (G x, y) of the corresponding graph GL specialized to xy = 1. We point out a relation between the Yamada polynomial, which is a colored Jones polynomial for trivalent graphs, and the chromatic polynomial, which is the Tutte polynomial specialized to the real axis. Given any graph G, the chromatic polynomial χG(k) is a polynomial in k such that when k is a positive integer, then χG(k) is the number of k-colorings of vertices of G such that no two vertices connected by an edge are given the same color. If the graph G is planar and trivalent, then the colored Jones polynomial for G with each edge colored by p2 is a Laurent polynomial in the loop variable d. This polynomial is called the Yamada polynomial, denoted as YG(d). Theorem 1.33. If G is a planar graph and ˆ its dual, then d−V χG(d2) = Y ˆ (d) where V = (# vertices of G) = (# faces of ˆ). For a proof, see [FFNWW]. 1.7. Yang-Baxter equation We will use the unitary Jones representations of braids to do quantum compu- tation. In the quantum circuit model (QCM) of quantum computation, the com- putational space is a tensor product of qudits, manifesting the locality of quantum mechanics explicitly. In TQC, the lack of natural tensor decompositions makes the topological model inconvenient for implementing QCM algorithms. It is desirable to have unitary solutions of the Yang-Baxter equation R: V ⊗ V → V ⊗ V so we can have a quantum computational model based on braiding with obvious locality. We have a choice for qudits, either V or V ⊗ V , but both run into problems: we don’t have good realizations of the Yang-Baxter space V in real materials, and no unitary solutions of the Yang-Baxter equation are known to the author such that the resulting representations of the braid groups have infinite images modulo phases. There is clearly a tension between unitarity and locality for braid group representations. The first nontrivial unitary solution [D] is the Bell matrix R = 1 √ 2 Ü 1 0 0 1 0 1 −1 0 0 1 1 0 −1 0 0 1 ê

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