### Topological Quantum Information Theory

"... This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recou ..."

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This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the

### ON PICTURE (2+1)-TQFTS

, 806

"... Dedicated to the memory of Xiao-Song Lin Abstract. The goal of the paper is an exposition of the simplest (2 + 1)-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and ..."

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Dedicated to the memory of Xiao-Song Lin Abstract. The goal of the paper is an exposition of the simplest (2 + 1)-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and should be sent to the 4th author. 1.

### Explorations in Dirac Fermions and Spin Liquids A dissertation presented by

, 2011

"... A significant portion of this dissertation is devoted to the study of the effects of impurities in substances whose low energy modes can be described by fermions obeying the gapless Dirac equation in 2+1 dimensions. First, we examine the case of a spin vacancy in the staggered flux spin liquid whose ..."

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A significant portion of this dissertation is devoted to the study of the effects of impurities in substances whose low energy modes can be described by fermions obeying the gapless Dirac equation in 2+1 dimensions. First, we examine the case of a spin vacancy in the staggered flux spin liquid whose excitations are Dirac fermions coupled to a U(1) gauge field. This vacancy leads to an anomalous Curie susceptibility and does not induce any local orders. Next, a Coulomb charge impurity placed on clean graphene is considered. We find that the Dirac quasiparticles in graphene do not screen the impurity charge, to all orders in perturbation theory. However, electronic correlations are found to induce a cloud of charge having the same sign as the impurity charge. We also analyze the case of a local impurity in graphene in the presence of a magnetic field and derive the spatial fourier transform of tunneling spectroscopy data obtained on an almost-clean

### From Quantum to Emergent Gravity: Theory and Phenomenology

, 2007

"... Let us assume that gravity is an emergent low-energy phenomenon arising from a topologically stable defect in momentum space – the Fermi point. What are the consequences? We discuss the natural values of fermion masses and cosmological constant; flatness of the Universe; bounds on Lorentz violation; ..."

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Let us assume that gravity is an emergent low-energy phenomenon arising from a topologically stable defect in momentum space – the Fermi point. What are the consequences? We discuss the natural values of fermion masses and cosmological constant; flatness of the Universe; bounds on Lorentz violation; etc.

### Spin Networks and Anyonic Topological Computing II

, 707

"... We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. ..."

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We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work.