Numerical Analysis of Quasiholes of the Moore-Read Wavefunction

We demonstrate numerically that non-Abelian quasihole excitations of the $\nu = 5/2$ fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the quasihole spacing is increased, the unitary transformation which describes winding two quasiholes around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four quasiholes become exponentially degenerate. We calculate the length scales for these two decays to be $\xi_{U} \approx 2.7 \ell_0$ and $\xi_{E} \approx 2.3 \ell_0$ respectively. Additionally we determine which fusion channel is lower in energy when two quasiholes are brought close together.Comment: 4 pages, 3 figure

Resources Required for Topological Quantum Factoring

We consider a hypothetical topological quantum computer where the qubits are comprised of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits (space) necessary to execute the most computationally expensive step of Shor's algorithm, modular exponentiation. For Ising anyons, we apply Bravyi's distillation method [S. Bravyi, Phys. Rev. A 73, 042313 (2006)] which combines topological and non-topological operations to allow for universal quantum computation. With reasonable restrictions on the physical parameters we find that factoring a 128 bit number requires approximately 10^3 Fibonacci anyons versus at least 3 x 10^9 Ising anyons. Other distillation algorithms could reduce the resources for Ising anyons substantially.Comment: 4+epsilon pages, 4 figure

Coulomb drag at \nu = 1/2: Composite fermion pairing fluctuations

We consider the Coulomb drag between two two-dimensional electron layers at filling factor \nu = 1/2 each, using a strong coupling approach within the composite fermion picture. Due to an attractive interlayer interaction, composite fermions are expected to form a paired state below a critical temperature T_c. We find that above T_c pairing fluctuations make the longitudinal transresistivity \rho_D increase with decreasing temperature. The pairing mechanism we study is very sensitive to density variations in the two layers, and to an applied current. We discuss possible relation to an experiment by Lilly et al. [Phys. Rev. Lett. 80, 1714 (1998)].Comment: REVTeX, 4 pages, 1 figur

Spin-pairing instabilities at the coincidence of two Landau levels

The effect of interactions near the coincidence of two Landau levels with opposite spins at filling factor 1/2 is investigated. By mapping to Composite Fermions it is shown that the fluctuations of the gauge field induces an effective attractive Fermion interaction. This can lead to a spin-singlet ground state that is separated from the excited states by a gap. The magnitude of the gap is evaluated. The results are consistent with the recently observed half-polarized states in the FQHE at a fixed filling factor. It is suggested that similar anomalies exist for other spin configurations in degenerate spin-up and spin-down Landau levels. An experiment for testing the spin-singlet state is proposed.Comment: to be published in Physical Review

Topological Quantum Computing with Only One Mobile Quasiparticle

In a topological quantum computer, universal quantum computation is performed by dragging quasiparticle excitations of certain two dimensional systems around each other to form braids of their world lines in 2+1 dimensional space-time. In this paper we show that any such quantum computation that can be done by braiding $n$ identical quasiparticles can also be done by moving a single quasiparticle around n-1 other identical quasiparticles whose positions remain fixed.Comment: 4 pages, 5 figure

Coulomb drag as a signature of the paired quantum Hall state

Motivated by the recent Coulomb drag experiment of M. P. Lilly et. al, we study the Coulomb drag in a two-layer system with Landau level filling factor $\nu=1/2$. We find that the drag conductivity in the incompressible paired quantum Hall state at zero temperature can be finite. The drag conductivity is also greatly enhanced above $T_c$, at which the transition between the weakly coupled compressible liquids and the paired quantum Hall liquid takes place. We discuss the implications of our results for the recent experiment.Comment: 4 pages, 1 figure included, replaced by the published versio

Braid Topologies for Quantum Computation

In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particle-like excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world-lines in three dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world-lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.Comment: 4 pages, 4 figures, minor revision

Topological Quantum Compiling

A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.Comment: 20 pages, 20 figures, published versio

Universal Leakage Elimination

``Leakage'' errors are particularly serious errors which couple states within a code subspace to states outside of that subspace thus destroying the error protection benefit afforded by an encoded state. We generalize an earlier method for producing leakage elimination decoupling operations and examine the effects of the leakage eliminating operations on decoherence-free or noiseless subsystems which encode one logical, or protected qubit into three or four qubits. We find that by eliminating the large class of leakage errors, under some circumstances, we can create the conditions for a decoherence free evolution. In other cases we identify a combination decoherence-free and quantum error correcting code which could eliminate errors in solid-state qubits with anisotropic exchange interaction Hamiltonians and enable universal quantum computing with only these interactions.Comment: 14 pages, no figures, new version has references updated/fixe

The J_1-J_2 antiferromagnet with Dzyaloshinskii-Moriya interaction on the square lattice: An exact diagonalization study

We examine the influence of an anisotropic interaction term of Dzyaloshinskii-Moriya (DM) type on the groundstate ordering of the J_1-J_2 spin-1/2-Heisenberg antiferromagnet on the square lattice. For the DM term we consider several symmetries corresponding to different crystal structures. For the pure J_1-J_2 model there are strong indications for a quantum spin liquid in the region of 0.4 < J_2/J_1 < 0.65. We find that a DM interaction influences the breakdown of the conventional antiferromagnetic order by i) shifting the spin liquid region, ii) changing the isotropic character of the groundstate towards anisotropic correlations and iii) creating for certain symmetries a net ferromagnetic moment.Comment: 7 pages, RevTeX, 6 ps-figures, to appear in J. Phys.: Cond. Ma