28,235 research outputs found
A class of quantum many-body states that can be efficiently simulated
We introduce the multi-scale entanglement renormalization ansatz (MERA), an
efficient representation of certain quantum many-body states on a D-dimensional
lattice. Equivalent to a quantum circuit with logarithmic depth and distinctive
causal structure, the MERA allows for an exact evaluation of local expectation
values. It is also the structure underlying entanglement renormalization, a
coarse-graining scheme for quantum systems on a lattice that is focused on
preserving entanglement.Comment: 4 pages, 5 figure
Entanglement renormalization
In the context of real-space renormalization group methods, we propose a
novel scheme for quantum systems defined on a D-dimensional lattice. It is
based on a coarse-graining transformation that attempts to reduce the amount of
entanglement of a block of lattice sites before truncating its Hilbert space.
Numerical simulations involving the ground state of a 1D system at criticality
show that the resulting coarse-grained site requires a Hilbert space dimension
that does not grow with successive rescaling transformations. As a result we
can address, in a quasi-exact way, tens of thousands of quantum spins with a
computational effort that scales logarithmically in the system's size. The
calculations unveil that ground state entanglement in extended quantum systems
is organized in layers corresponding to different length scales. At a quantum
critical point, each rellevant length scale makes an equivalent contribution to
the entanglement of a block with the rest of the system.Comment: 4 pages, 4 figures, updated versio
Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit
Subspace clustering methods based on , or nuclear norm
regularization have become very popular due to their simplicity, theoretical
guarantees and empirical success. However, the choice of the regularizer can
greatly impact both theory and practice. For instance, regularization
is guaranteed to give a subspace-preserving affinity (i.e., there are no
connections between points from different subspaces) under broad conditions
(e.g., arbitrary subspaces and corrupted data). However, it requires solving a
large scale convex optimization problem. On the other hand, and
nuclear norm regularization provide efficient closed form solutions, but
require very strong assumptions to guarantee a subspace-preserving affinity,
e.g., independent subspaces and uncorrupted data. In this paper we study a
subspace clustering method based on orthogonal matching pursuit. We show that
the method is both computationally efficient and guaranteed to give a
subspace-preserving affinity under broad conditions. Experiments on synthetic
data verify our theoretical analysis, and applications in handwritten digit and
face clustering show that our approach achieves the best trade off between
accuracy and efficiency.Comment: 13 pages, 1 figure, 2 tables. Accepted to CVPR 2016 as an oral
presentatio
Creation and Manipulation of Anyons in the Kitaev Model
We analyze the effect of local spin operators in the Kitaev model on the
honeycomb lattice. We show, in perturbation around the isolated-dimer limit,
that they create Abelian anyons together with fermionic excitations which are
likely to play a role in experiments. We derive the explicit form of the
operators creating and moving Abelian anyons without creating fermions and show
that it involves multi-spin operations. Finally, the important experimental
constraints stemming from our results are discussed.Comment: 4 pages, 3 figures, published versio
Provable Self-Representation Based Outlier Detection in a Union of Subspaces
Many computer vision tasks involve processing large amounts of data
contaminated by outliers, which need to be detected and rejected. While outlier
detection methods based on robust statistics have existed for decades, only
recently have methods based on sparse and low-rank representation been
developed along with guarantees of correct outlier detection when the inliers
lie in one or more low-dimensional subspaces. This paper proposes a new outlier
detection method that combines tools from sparse representation with random
walks on a graph. By exploiting the property that data points can be expressed
as sparse linear combinations of each other, we obtain an asymmetric affinity
matrix among data points, which we use to construct a weighted directed graph.
By defining a suitable Markov Chain from this graph, we establish a connection
between inliers/outliers and essential/inessential states of the Markov chain,
which allows us to detect outliers by using random walks. We provide a
theoretical analysis that justifies the correctness of our method under
geometric and connectivity assumptions. Experimental results on image databases
demonstrate its superiority with respect to state-of-the-art sparse and
low-rank outlier detection methods.Comment: 16 pages. CVPR 2017 spotlight oral presentatio
Characterizing topological order by studying the ground states of an infinite cylinder
Given a microscopic lattice Hamiltonian for a topologically ordered phase, we
describe a tensor network approach to characterize its emergent anyon model
and, in a chiral phase, also its gapless edge theory. First, a tensor network
representation of a complete, orthonormal set of ground states on a cylinder of
infinite length and finite width is obtained through numerical optimization.
Each of these ground states is argued to have a different anyonic flux
threading through the cylinder. In a chiral phase, the entanglement spectrum of
each ground state is seen to reveal a different sector of the corresponding
gapless edge theory. A quasi-orthogonal basis on the torus is then produced by
chopping off and reconnecting the tensor network representation on the
cylinder. Elaborating on the recent proposal of [Y. Zhang et al. Phys. Rev. B
85, 235151 (2012)], a rotation on the torus yields an alternative basis of
ground states and, through the computation of overlaps between bases, the
modular matrices S and U (containing the mutual and self statistics of the
different anyon species) are extracted. As an application, we study the
hard-core boson Haldane model by using the two-dimensional density matrix
renormalization group. A thorough characterization of the universal properties
of this lattice model, both in the bulk and at the edge, unambiguously shows
that its ground space realizes the \nu=1/2 bosonic Laughlin state.Comment: 10 pages, 11 figure
Simulation of anyons with tensor network algorithms
Interacting systems of anyons pose a unique challenge to condensed matter
simulations due to their non-trivial exchange statistics. These systems are of
great interest as they have the potential for robust universal quantum
computation, but numerical tools for studying them are as yet limited. We show
how existing tensor network algorithms may be adapted for use with systems of
anyons, and demonstrate this process for the 1-D Multi-scale Entanglement
Renormalisation Ansatz (MERA). We apply the MERA to infinite chains of
interacting Fibonacci anyons, computing their scaling dimensions and local
scaling operators. The scaling dimensions obtained are seen to be in agreement
with conformal field theory. The techniques developed are applicable to any
tensor network algorithm, and the ability to adapt these ansaetze for use on
anyonic systems opens the door for numerical simulation of large systems of
free and interacting anyons in one and two dimensions.Comment: Fixed typos, matches published version. 16 pages, 21 figures, 4
tables, RevTeX 4-1. For a related work, see arXiv:1006.247
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