Universal Quantum Computation with the nu=5/2 Fractional Quantum Hall State

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the Fractional Quantum Hall Effect state at Landau level filling fraction nu=5/2. Since the braid group representation describing statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy non-topological operations such as direct short-range interaction between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for non-topological operations is above 14%. The total number of non-topological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\log L)^3$.Comment: 17 pages, 12 eps figure

Topological entanglement entropy

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator \rho for the degrees of freedom in the interior. The von Neumann entropy S(\rho) of this density operator, a measure of the entanglement of the interior and exterior variables, has the form S(\rho)= \alpha L -\gamma + ..., where the ellipsis represents terms that vanish in the limit L\to\infty. The coefficient \alpha, arising from short wavelength modes localized near the boundary, is nonuniversal and ultraviolet divergent, but -\gamma is a universal additive constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for \gamma in terms of properties of the superselection sectors of the medium.Comment: 4 pages, 3 eps figures. v2: reference adde

Exact results for spin dynamics and fractionization in the Kitaev Model

We present certain exact analytical results for dynamical spin correlation functions in the Kitaev Model. It is the first result of its kind in non-trivial quantum spin models. The result is also novel: in spite of presence of gapless propagating Majorana fermion excitations, dynamical two spin correlation functions are identically zero beyond nearest neighbor separation, showing existence of a gapless but short range spin liquid. An unusual, \emph{all energy scale fractionization}of a spin -flip quanta, into two infinitely massive $\pi$-fluxes and a dynamical Majorana fermion, is shown to occur. As the Kitaev Model exemplifies topological quantum computation, our result presents new insights into qubit dynamics and generation of topological excitations.Comment: 4 pages, 2 figures. Typose corrected, figure made better, clarifying statements and references adde

Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

An Isomonodromy Cluster of Two Regular Singularities

We consider a linear $2\times2$ matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as $P_6\to P_5$, where $P_n$ is the $n$-th Painlev\'e equation. We also discuss some general problems which arise while studying the above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure

A comprehensive introduction to the theory of word-representable graphs

Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy⋯ (of even or odd length) or a word yxyx⋯  (of even or odd length). A graph G=(V,E) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy ∈ E.   Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-representable graphs including the most recent developments in the area

Constructing Functional Braids for Low-Leakage Topological Quantum Computing

We discuss how to significantly reduce leakage errors in topological quantum computation by introducing an irrelevant error in phase, using the construction of a CNOT gate in the Fibonacci anyon model as a concrete example. To be specific, we construct a functional braid in a six-anyon Hilbert space that exchanges two neighboring anyons while conserving the encoded quantum information. The leakage error is $\sim$$10^{-10}$ for a braid of $\sim$100 interchanges of anyons. Applying the braid greatly reduces the leakage error in the construction of generic controlled-rotation gates.Comment: 5 pages, 4 figures, updated, accepeted by Phys. Rev.

Disordered Topological Insulators via C∗C^*-Algebras

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z_2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z_2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.Comment: Added detail regarding the mapping of almost commuting unitary matrices to almost commuting Hermitian matrices that form an approximate representation of the sphere. 6 pages, 6 figure

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlev\'{e} Equation. I

The degenerate third Painlev\'{e} equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}{\tau} + \frac{1}{\tau}(-8 \epsilon u^{2} + 2ab) + \frac{b^{2}}{u}$, where $\epsilon,b \in \mathbb{R}$, and $a \in \mathbb{C}$, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as $\tau \to \pm 0$ and $\pm i0$ solution and general regular as $\tau \to \pm \infty$ and $\pm i \infty$ solution are obtained.Comment: 40 pages, LaTeX2

Universal Quantum Computation with ideal Clifford gates and noisy ancillas

We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state $|0>$, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $\rho$, which should be regarded as a parameter of the model. Our goal is to determine for which $\rho$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $\rho$ and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain ``magic'' directions. If the polarization of $\rho$ along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.Comment: 15 pages, 4 figures, revtex