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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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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],
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 30 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
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
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 14 (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.
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 13 (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,
A differential ideal of symmetric polynomials spanned by Jack polynomials at β
 r − 1)/(k
, 2002
"... Abstract. For each pair of positive integers (k, r) such that k+1, r −1 are coprime, we introduce an ideal I (k,r) n of the ring of symmetric polynomials C[x1, · · · , xn] Sn. The ideal I (k,r) n has a basis consisting of Jack polynomials with parameter β = −(r − 1)/(k + 1), and admits an action ..."
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Cited by 11 (5 self)
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Abstract. For each pair of positive integers (k, r) such that k+1, r −1 are coprime, we introduce an ideal I (k,r) n of the ring of symmetric polynomials C[x1, · · · , xn] Sn. The ideal I (k,r) n has a basis consisting of Jack polynomials with parameter β = −(r − 1)/(k + 1), and admits an action of a family of differential operators of Dunkl type including the positive half of the Virasoro algebra. The space I (k,2) n coincides with the space of all symmetric polynomials in n variables which vanish when k + 1 variables are set equal. The space I (2,r) n coincides with the space of correlation functions of an abelian current of a vertex operator algebra related to Virasoro minimal series (3, r + 2). 1.
Rapidly rotating atomic gases
 Advances in Physics, 57:539–616
, 2008
"... This article reviews developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single component atomic Bose gas, which (at least at rest) forms a BoseEinstein condensate. Rotation leads to the formation of quantized vortices which o ..."
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This article reviews developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single component atomic Bose gas, which (at least at rest) forms a BoseEinstein condensate. Rotation leads to the formation of quantized vortices which order into a vortex array, in close analogy with the behaviour of superfluid helium. Under conditions of rapid rotation, when the vortex density becomes large, atomic Bose gases offer the possibility to explore the physics of quantized vortices in novel parameter regimes. First, there is an interesting regime in which the vortices become sufficiently dense that their cores – as set by the healing length – start to overlap. In this regime, the theoretical description simplifies, allowing a reduction to single particle states in the lowest Landau level. Second, one can envisage entering a regime of very high vortex density, when the number of vortices becomes comparable to the number of particles in the gas. In this regime, theory predicts the appearance of a series of strongly correlated phases, which can be viewed as bosonic versions of fractional quantum Hall states. This article describes the
Wavefunctions for topological quantum registers
 Ann. Phys. (N. Y
"... We present explicit wavefunctions for quasihole excitations over a variety of nonabelian quantum Hall states: the ReadRezayi states with k ≥ 3 clustering properties and a paired spinsinglet quantum Hall state. Quasiholes over these states constitute a topological quantum register, which can be a ..."
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We present explicit wavefunctions for quasihole excitations over a variety of nonabelian quantum Hall states: the ReadRezayi states with k ≥ 3 clustering properties and a paired spinsinglet quantum Hall state. Quasiholes over these states constitute a topological quantum register, which can be addressed by braiding quasiholes. We obtain the braid properties by direct inspection of the quasihole wavefunctions. We establish that the braid properties for the paired spinsinglet state are those of ‘Fibonacci anyons’, and thus suitable for universal quantum computation. Our derivations in this paper rely on explicit computations in the parafermionic Conformal Field Theories that underly these particular quantum Hall states. 1
NonAbelian Anyons and Topological Quantum Computation
, 2007
"... 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 know ..."
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Cited by 7 (1 self)
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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, which
NonAbelian quantum Hall states – exclusion statistics, Kmatrices and duality
, 2000
"... We study excitations in edge theories for nonabelian quantum Hall states, focussing on the spin polarized states proposed by Read and Rezayi and on the spin singlet states proposed by two of the authors. By studying the exclusion statistics properties of edgeelectrons and edgequasiholes, we arr ..."
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Cited by 6 (3 self)
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We study excitations in edge theories for nonabelian quantum Hall states, focussing on the spin polarized states proposed by Read and Rezayi and on the spin singlet states proposed by two of the authors. By studying the exclusion statistics properties of edgeelectrons and edgequasiholes, we arrive at a novel Kmatrix structure. Interestingly, the duality between the electron and quasihole sectors links the pseudoparticles that are characteristic for nonabelian statistics with composite particles that are associated to the ‘pairing physics’ of the nonabelian quantum Hall states.