30,113 research outputs found
Measuring Majorana fermions qubit state and non-Abelian braiding statistics in quenched inhomogeneous spin ladders
We study the Majorana fermions (MFs) in a spin ladder model. We propose and
numerically show that the MFs qubit state can be read out by measuring the
fusion excitation in the quenched inhomogeneous spin ladders. Moreover, we
construct an exactly solvable T-junction spin ladder model, which can be used
to implement braiding operations of MFs. With the braiding processes simulated
numerically as non-equilibrium quench processes, we verify that the MFs in our
spin ladder model obey the non-Abelian braiding statistics. Our scheme not only
provides a promising platform to study the exotic properties of MFs, but also
has broad range of applications in topological quantum computation.Comment: 5+3 pages, 6 figure
Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time
We present the first almost-linear time algorithm for constructing
linear-sized spectral sparsification for graphs. This improves all previous
constructions of linear-sized spectral sparsification, which requires
time.
A key ingredient in our algorithm is a novel combination of two techniques
used in literature for constructing spectral sparsification: Random sampling by
effective resistance, and adaptive constructions based on barrier functions.Comment: 22 pages. A preliminary version of this paper is to appear in
proceedings of the 56th Annual IEEE Symposium on Foundations of Computer
Science (FOCS 2015
An SDP-Based Algorithm for Linear-Sized Spectral Sparsification
For any undirected and weighted graph with vertices and
edges, we call a sparse subgraph of , with proper reweighting of the
edges, a -spectral sparsifier if holds for any , where and
are the respective Laplacian matrices of and . Noticing that
time is needed for any algorithm to construct a spectral sparsifier and a
spectral sparsifier of requires edges, a natural question is to
investigate, for any constant , if a -spectral
sparsifier of with edges can be constructed in time,
where the notation suppresses polylogarithmic factors. All previous
constructions on spectral sparsification require either super-linear number of
edges or time.
In this work we answer this question affirmatively by presenting an algorithm
that, for any undirected graph and , outputs a
-spectral sparsifier of with edges in
time. Our algorithm is based on three novel
techniques: (1) a new potential function which is much easier to compute yet
has similar guarantees as the potential functions used in previous references;
(2) an efficient reduction from a two-sided spectral sparsifier to a one-sided
spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a
semi-definite program.Comment: To appear at STOC'1
Distinct Spin Liquids and their Transitions in Spin-1/2 XXZ Kagome Antiferromagnets
By using the density matrix renormalization group, we study the spin-liquid
phases of spin- XXZ kagome antiferromagnets. We find that the emergence of
spin liquid phase does not depend on the anisotropy of the XXZ interaction. In
particular, the two extreme limits---Ising (strong interaction) and XY
(zero interaction)---host the same spin-liquid phases as the isotropic
Heisenberg model. Both the time-reversal-invariant spin liquid and the chiral
spin liquid with spontaneous time-reversal symmetry breaking are obtained. We
show they evolve continuously into each other by tuning the second- and
third-neighbor interactions. At last, we discuss the possible implication of
our results on the nature of spin liquid in nearest neighbor XXZ kagome
antiferromagnets, including the most studied nearest neighbor spin- kagome
anti-ferromagnetic Heisenberg model
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