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Topological quantum computation
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations ..."
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Cited by 109 (14 self)
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Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2Dmagnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e−αℓ, where ℓ is a length scale, and α is some positive constant. In contrast, the “presumptive ” qubitmodel of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10−4) before computation can be stabilized. Quantum computation is a catchall for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19],
A magnetic model with a possible ChernSimons phase
 Commun. Math. Phys
"... A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a co ..."
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Cited by 27 (3 self)
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A rather elementary family of local Hamiltonians H◦,ℓ,ℓ = 1,2,3,..., is described for a 2−dimensional quantum mechanical system of spin = 1 2 particles. On the torus, the ground state space G◦,ℓ is essentially infinite dimensional but may collapse under “perturbation ” to an anyonic system with a complete mathematical description: the quantum double of the SO(3)−ChernSimons modular functor at q = e 2πi/ℓ+2 which we call DEℓ. The Hamiltonian H◦,ℓ defines a quantum loop gas. We argue that for ℓ = 1 and 2, G◦,ℓ is unstable and the collapse to Gǫ,ℓ ∼ = DEℓ can occur truly by perturbation. For ℓ ≥ 3 G◦,ℓ is stable and in this case finding Gǫ,ℓ ∼ = DEℓ must require either ǫ> ǫℓ> 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state G◦,ℓ of H◦,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gǫ,ℓ described by a quotient algebra. By classification, this implies Gǫ,ℓ ∼ = DEℓ. The fundamental point is that nonlinear structures
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
"... We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general famil ..."
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Cited by 24 (5 self)
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We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the wellknown plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithms on a local qubit implementation of the unitary JonesWenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT twovariable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a selfcontained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. This theorem was originally proved by Freedman, Larsen and Wang in the context of topological quantum computation, and the necessary notion of approximation was later provided by Bordewich et al. Our proof is simpler as it uses a more natural encoding of twoqubit unitaries into the rectangular representation of the eightstrand braid group. We then give QCMAcomplete and PSPACEcomplete problems which are based on braids. Finally, we conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #Phard problem, while learning its most significant bit is PPhard, without taking the usual route through the Tutte polynomial and graph coloring. 1
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 13 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Extraspecial 2groups and images of braid group representations
 J. Knot Theory Ramifications
"... Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompo ..."
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Cited by 13 (6 self)
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Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the wellknown Jones representations of Bn factoring over TemperleyLieb algebras and the corresponding link invariants. 1.
Nonabelian anyons and topological quantum computation
 Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 10 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
From Quantum Groups to Unitary Modular Tensor Categories
 CONTEMPORARY MATHEMATICS
"... Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently propos ..."
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Cited by 8 (6 self)
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Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.
NonAbelian Anyons and Topological Quantum Computation. arxiv: condmat.strel/0707.1889
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 7 (1 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations ..."
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Cited by 5 (3 self)
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Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F. 1.
Quantum Morphing and the Jones Polynomial
, 2001
"... We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the deg ..."
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Cited by 5 (2 self)
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We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the degree of the roots of unit is getting larger, the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the Jones polynomial. We call such a phenomenon "quantum morphing". Evaluations of the Jones polynomial at roots of unity play a crucial role, for example in the volume conjecture.