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13
A modular functor which is universal for quantum computation
 Comm. Math. Phys
"... Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based o ..."
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Cited by 84 (17 self)
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Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere. 1.
Representation Theory of ChernSimons Observables
, 1995
"... In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional surfac ..."
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Cited by 33 (0 self)
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In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.
Combinatorial Quantization of the Hamiltonian ChernSimons Theory
, 1994
"... Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian ChernSimons theory. We introduce the ChernSimons theory on the lattice which reproduces the results of the continuous theory exactly. The lattice model enjoys the symmetry with re ..."
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Cited by 32 (3 self)
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Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian ChernSimons theory. We introduce the ChernSimons theory on the lattice which reproduces the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian ChernSimons theory equipped with a *operation and a positive inner product.
Geometrical measurements in threedimensional quantum gravity
 J. Phys. A: Math. Gen. 18 Suppl
"... A set of observables is described for the topological quantum field theory which describes quantum gravity in three spacetime dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which hav ..."
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Cited by 20 (5 self)
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A set of observables is described for the topological quantum field theory which describes quantum gravity in three spacetime dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform. 1
Lattice topological field theory on nonorientable surfaces, J.Math
 Phys
, 1997
"... The lattice definition of the twodimensional topological quantum field theory [Fukuma, et al, Commun. Math. Phys. 161, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a onetoone correspondence between real associative ∗algebras and ..."
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Cited by 9 (0 self)
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The lattice definition of the twodimensional topological quantum field theory [Fukuma, et al, Commun. Math. Phys. 161, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a onetoone correspondence between real associative ∗algebras and the topological state sum invariants defined on such surfaces. The partition and npoint functions on all twodimensional surfaces (connected sums of the Klein bottle or projective plane and gtori) are defined and computed for arbitrary ∗algebras in general, and for the the group ring A = IR[G] of discrete groups G, in particular. 1 1
The PonzanoRegge model
, 2008
"... The definition of the PonzanoRegge statesum model of threedimensional quantum gravity with a class of local observables is developed. The main definition of the PonzanoRegge model in this paper is determined by its reformulation in terms of group variables. The regularisation is defined and a pro ..."
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Cited by 5 (1 self)
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The definition of the PonzanoRegge statesum model of threedimensional quantum gravity with a class of local observables is developed. The main definition of the PonzanoRegge model in this paper is determined by its reformulation in terms of group variables. The regularisation is defined and a proof is given that the partition function is welldefined only when a certain cohomological criterion is satisfied. In that case, the partition function may be expressed in terms of a topological invariant, the Reidemeister torsion. This proves the independence of the definition on the triangulation of the 3manifold and on those arbitrary choices made in the regularisation. A further corollary is that when the observable is a knot, the partition function (when it exists) can be written in terms of the Alexander polynomial of the knot. Various examples of observables in S 3 are computed explicitly. Alternative regularisations of the PonzanoRegge model by the simple cutoff procedure and by the limit of the TuraevViro model are discussed, giving successes and limitations of these approaches. 1
Observables in 3dimensional quantum gravity and topological invariants. grqc/0401093 Class.Quant.Grav
, 2004
"... Abstract. In this paper we report some results on the expectation values of a set of observales introduced for 3dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the TuraevViro model. Instead of giving a formal description of the obsevables, we jus ..."
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Cited by 2 (1 self)
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Abstract. In this paper we report some results on the expectation values of a set of observales introduced for 3dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the TuraevViro model. Instead of giving a formal description of the obsevables, we just formulate the paper by examples. This means that we just show how an idea works with particular cases and give a way to compute ’expectation values ’ in general by a topological procedure. 1
A Lattice Model of Local Algebras of Observables and Fields With Braid Group Statistics
, 1995
"... Using the 6jsymbols and the Rmatrix for the quantum group Sl q (2; C) at roots of unity we construct local algebras of observables and fields with braid group statistics on the lattice Z. These algebras are closely related to the XXZHeisenberg model and the RSOS models thus exhibiting the qua ..."
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Cited by 1 (0 self)
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Using the 6jsymbols and the Rmatrix for the quantum group Sl q (2; C) at roots of unity we construct local algebras of observables and fields with braid group statistics on the lattice Z. These algebras are closely related to the XXZHeisenberg model and the RSOS models thus exhibiting the quantum group symmetry of these models. Our discussion relates the theory of integrable lattice models to the DoplicherHaag Roberts theory of superselection sectors. The construction of these algebras is a variant of the path space construction of Ocneanu and Sunders which replaces the usual tensor product construction of lattice models in statistical mechanics and extends previous discussions by Pasquier. Our construction is based on the theory of coloured graphs on S 2 and the associated WignerEckhart theorem obtained previously by the authors. 1 Introduction Lattice models of quantum statistical models are usually based on the concept of the tensor product and variants thereof like the ...
Invariants of 3Manifolds, Unitary Representations of the Mapping Class Group and Numerical Calculations
, 1996
"... Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed 3manifolds by Turaev and Viro as state sums in terms of quantum 6jsymbols for Sl q (2; C) at roots of unity (q = exp ßi=r) leads to the unitary representation of the ..."
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Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed 3manifolds by Turaev and Viro as state sums in terms of quantum 6jsymbols for Sl q (2; C) at roots of unity (q = exp ßi=r) leads to the unitary representation of the mapping class group found by Moore and Seiberg. Via a Heegaard decomposition this invariant may therefore be written as the absolute square of a certain matrix element of a suitable group element in this representation. For an arbitrary Dehn surgery on a figureeight knot we provide an explicit form for this matrix element involving just one 6jsymbol. This expression is analyzed numerically and compared with the conjectured large k = r \Gamma 2 asymptotics of the ChernSimonsWitten state sum [1], whose absolute square is the TuraevViro state sum. In particular we find numerical agreement concerning the values of the ChernSimons invariants for the flat SU(2) connections as predicted by ...
A quantum group version of quantum gauge theories in two dimensions
, 1992
"... For the special case of the quantum group SLq(2,C) (q = exp πi/r, r ≥ 3) we present an alternative approach to quantum gauge theories in two dimensions. We exhibit the similarities to Witten’s combinatorial approach which is based on ideas of Migdal. The main ingredient is the TuraevViro combinator ..."
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For the special case of the quantum group SLq(2,C) (q = exp πi/r, r ≥ 3) we present an alternative approach to quantum gauge theories in two dimensions. We exhibit the similarities to Witten’s combinatorial approach which is based on ideas of Migdal. The main ingredient is the TuraevViro combinatorial construction of topological invariants of closed, compact 3manifolds and its extension to arbitrary compact 3manifolds as given by the authors in collaboration with W. Müller.