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

Towards Universal Topological Quantum Computation in the ν=5/2\nu=5/2 Fractional Quantum Hall State

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\nu = 5/2$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry.Comment: A related paper, cond-mat/0512072, explains the topological issues in greater detail. It may help the reader to look at this alternate presentation if confused about any poin

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

On a suggestion relating topological and quantum mechanical entanglements

We analyze a recent suggestion \cite{kauffman1,kauffman2} on a possible relation between topological and quantum mechanical entanglements. We show that a one to one correspondence does not exist, neither between topologically linked diagrams and entangled states, nor between braid operators and quantum entanglers. We also add a new dimension to the question of entangling properties of unitary operators in general.Comment: RevTex, 7 eps figures, to be published in Phys. Lett. A (2004

COBE ground segment attitude determination

The Cosmic Background Explorer (COBE) spacecraft was launched in November 1989 by NASA to survey the sky for primordial radiation left from the Big Bang explosion. The success of the mission requires an accurate determination of the spacecraft attitude. While the accuracy of the attitude obtained from the attitude sensors is adequate for two of the experiments, the higher accuracy required by the Diffuse Infrared Background Experiment (DIRBE) is obtained by using the DIRBE instrument as a special type of star sensor. Presented here is an overview of the attitude processing algorithms used at the Cosmology Data Analysis Center (CDAC) and the results obtained from the flight data

Group Approach to Quantization of Yang-Mills Theories: A Cohomological Origin of Mass

New clues for the best understanding of the nature of the symmetry-breaking mechanism are revealed in this paper. A revision of the standard gauge transformation properties of Yang-Mills fields, according to a group approach to quantization scheme, enables the gauge group coordinates to acquire dynamical content outside the null mass shell. The corresponding extra (internal) field degrees of freedom are transferred to the vector potentials to conform massive vector bosons.Comment: 21 pages, LaTeX, no figures; final for

The Three Loop Equation of State of QED at High Temperature

We present the three loop contribution (order $e^4$) to the pressure of massless quantum electrodynamics at nonzero temperature. The calculation is performed within the imaginary time formalism. Dimensional regularization is used to handle the usual, intermediate stage, ultraviolet and infrared singularities, and also to prevent overcounting of diagrams during resummation.Comment: ANL-HEP-PR-94-02, SPhT/94-054 (revised final version

Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at Landau level filling fraction $\nu=5/2$. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the $\nu=5/2$ state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically-protected qubit on which a logical NOT operation is performed by quasiparticle braiding. Using the measured excitation gap at $\nu=5/2$, we estimate the error rate to be $10^{-30}$ or lower

COBE ground segment gyro calibration

Discussed here is the calibration of the scale factors and rate biases for the Cosmic Background Explorer (COBE) spacecraft gyroscopes, with the emphasis on the adaptation for COBE of an algorithm previously developed for the Solar Maximum Mission. Detailed choice of parameters, convergence, verification, and use of the algorithm in an environment where the reference attitudes are determined form the Sun, Earth, and star observations (via the Diffuse Infrared Background Experiment (DIRBE) are considered. Results of some recent experiments are given. These include tests where the gyro rate data are corrected for the effect of the gyro baseplate temperature on the spacecraft electronics