Large Fourier transforms never exactly realized by braiding conformal blocks

Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set \{\U(2), \textrm{CNOT}\}, the discrete Fourier transforms $F_N=(\omega^{ij})_{N\times N},i,j=0,1,..., N-1, \omega=e^{\frac{2\pi i}{N}}$, can be realized exactly by quantum circuits of size $O(n^2), n=\textrm{log}N$, and so can the discrete sine/cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci (2+1)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms $F_N$ and the discrete sine/cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that approximation is unavoidable to implement the Fourier transforms by braiding conformal blocks

Topological Phases: An Expedition off Lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of `baby universe', Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger's theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.Comment: 38 pages, 22 figure

Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates

We construct an exactly solvable Hamiltonian acting on a 3-dimensional lattice of spin-$\frac 1 2$ systems that exhibits topological quantum order. The ground state is a string-net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends upon the homology of the 3-manifold. We generalize the system to $D\geq 4$, were different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.Comment: Revtex4 file, color figures, minor correction

Holographic Normal Ordering and Multi-particle States in the AdS/CFT Correspondence

The general correlator of composite operators of N=4 supersymmetric gauge field theory is divergent. We introduce a means for renormalizing these correlators by adding a boundary theory on the AdS space correcting for the divergences. Such renormalizations are not equivalent to the standard normal ordering of current algebras in two dimensions. The correlators contain contact terms that contribute to the OPE; we relate them diagrammatically to correlation functions of compound composite operators dual to multi-particle states.Comment: 18 pages, one equation corr., further comments and refs. adde

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

The Non-Trapping Degree of Scattering

We consider classical potential scattering. If no orbit is trapped at energy E, the Hamiltonian dynamics defines an integer-valued topological degree. This can be calculated explicitly and be used for symbolic dynamics of multi-obstacle scattering. If the potential is bounded, then in the non-trapping case the boundary of Hill's Region is empty or homeomorphic to a sphere. We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics defines an integer-valued topological degree deg(E) < 2. This is calculated explicitly for all potentials, and exactly the integers < 2 are shown to occur for suitable potentials. The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that the boundary of Hill's Region in configuration space is either empty or homeomorphic to a sphere. However, in many situations one can decompose a potential into a sum of non-trapping potentials with non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.Comment: 25 pages, 1 figure Revised and enlarged version, containing more detailed proofs and remark

The Equation of State for Dense QCD and Quark Stars

We calculate the equation of state for degenerate quark matter to leading order in hard-dense-loop (HDL) perturbation theory. We solve the Tolman-Oppenheimer-Volkov equations to obtain the mass-radius relation for dense quark stars. Both the perturbative QCD and the HDL equations of state have a large variation with respect to the renormalization scale for quark chemical potential below 1 GeV which leads to large theoretical uncertainties in the quark star mass-radius relation.Comment: 7 pages, 3 figure

Quantum Knitting

We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a `universal problem', namely the hardest problem that a quantum computer can efficiently handle.Comment: 29 pages, 5 figures; to appear in Laser Journa

Interacting anyons in topological quantum liquids: The golden chain

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (`identity') channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the two-dimensional tricritical Ising model is given using the restricted-solid-on-solid (RSOS) representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.Comment: 5 pages, 4 figure

A Field-theoretical Interpretation of the Holographic Renormalization Group

A quantum-field theoretical interpretation is given to the holographic RG equation by relating it to a field-theoretical local RG equation which determines how Weyl invariance is broken in a quantized field theory. Using this approach we determine the relation between the holographic C theorem and the C theorem in two-dimensional quantum field theory which relies on the Zamolodchikov metric. Similarly we discuss how in four dimensions the holographic C function is related to a conjectured field-theoretical C function. The scheme dependence of the holographic RG due to the possible presence of finite local counterterms is discussed in detail, as well as its implications for the holographic C function. We also discuss issues special to the situation when mass deformations are present. Furthermore we suggest that the holographic RG equation may also be obtained from a bulk diffeomorphism which reduces to a Weyl transformation on the boundary.Comment: 24 pages, LaTeX, no figures; references added, typos corrected, paragraph added to section