2,801 research outputs found
One-dimensional many-body entangled open quantum systems with tensor network methods
We present a collection of methods to simulate entangled dynamics of open
quantum systems governed by the Lindblad equation with tensor network methods.
Tensor network methods using matrix product states have been proven very useful
to simulate many-body quantum systems and have driven many innovations in
research. Since the matrix product state design is tailored for closed
one-dimensional systems governed by the Schr\"odinger equation, the next step
for many-body quantum dynamics is the simulation of open quantum systems. We
review the three dominant approaches to the simulation of open quantum systems
via the Lindblad master equation: quantum trajectories, matrix product density
operators, and locally purified tensor networks. Selected examples guide
possible applications of the methods and serve moreover as a benchmark between
the techniques. These examples include the finite temperature states of the
transverse quantum Ising model, the dynamics of an exciton traveling under the
influence of spontaneous emission and dephasing, and a double-well potential
simulated with the Bose-Hubbard model including dephasing. We analyze which
approach is favorable leading to the conclusion that a complete set of all
three methods is most beneficial, push- ing the limits of different scenarios.
The convergence studies using analytical results for macroscopic variables and
exact diagonalization methods as comparison, show, for example, that matrix
product density operators are favorable for the exciton problem in our study.
All three methods access the same library, i.e., the software package Open
Source Matrix Product States, allowing us to have a meaningful comparison
between the approaches based on the selected examples. For example, tensor
operations are accessed from the same subroutines and with the same
optimization eliminating one possible bias in a comparison of such numerical
methods.Comment: 24 pages, 8 figures. Small extension of time evolution section and
moving quantum simulators to introduction in comparison to v
Support vector machine for functional data classification
In many applications, input data are sampled functions taking their values in
infinite dimensional spaces rather than standard vectors. This fact has complex
consequences on data analysis algorithms that motivate modifications of them.
In fact most of the traditional data analysis tools for regression,
classification and clustering have been adapted to functional inputs under the
general name of functional Data Analysis (FDA). In this paper, we investigate
the use of Support Vector Machines (SVMs) for functional data analysis and we
focus on the problem of curves discrimination. SVMs are large margin classifier
tools based on implicit non linear mappings of the considered data into high
dimensional spaces thanks to kernels. We show how to define simple kernels that
take into account the unctional nature of the data and lead to consistent
classification. Experiments conducted on real world data emphasize the benefit
of taking into account some functional aspects of the problems.Comment: 13 page
From Nagaoka's ferromagnetism to flat-band ferromagnetism and beyond: An introduction to ferromagnetism in the Hubbard model
This is a self-contained review about ferromagnetism in the Hubbard model,
which should be accessible to readers with various backgrounds who are new to
the field.
We describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail,
giving all necessary backgrounds as well as complete (but elementary)
mathematical proofs. By studying an intermediate model called long-range
hopping model, we also demonstrate that there is indeed a deep relation between
these two seemingly different approaches to ferromagnetism.
We further discuss some attempts to go beyond these approaches. We briefly
discuss recent rigorous example of ferromagnetism in the Hubbard model which
has neither infinitely large parameters nor completely flat bands. We give
preliminary discussions about possible experimental realizations of the
(nearly-)flat-band ferromagnetism. Finally we focus on some theoretical
attempts to understand metallic ferromagnetism. We discuss three artificial
one-dimensional models in which the existence of metallic ferromagnetism can be
easily proved.Comment: LaTeX2e, 72 pages, 17 epsf figures. Many minor corrections made in
March 1998. This is the final version, which will appear in Prog. Theor.
Phys. 99 (invited paper
Supersymmetric Quantum Mechanics for String-Bits
We develop possible versions of supersymmetric single particle quantum
mechanics, with application to superstring-bit models in view. We focus
principally on space dimensions , the transverse dimensionalities of
superstring in space-time dimensions. These are the cases for which
``classical'' superstring makes sense, and also the values of for which
Hooke's force law is compatible with the simplest superparticle dynamics. The
basic question we address is: When is it possible to replace such harmonic
force laws with more general ones, including forces which vanish at large
distances? This is an important question because forces between string-bits
that do not fall off with distance will almost certainly destroy cluster
decomposition. We show that the answer is affirmative for , negative for
, and so far inconclusive for .Comment: 17 pages, Late
Attribute Relationship Analysis in Outlier Mining and Stream Processing
The main theme of this thesis is to unite two important fields of data analysis, outlier mining and attribute relationship analysis. In this work we establish the connection between these two fields. We present techniques which exploit this connection, allowing to improve outlier detection in high dimensional data. In the second part of the thesis we extend our work to the emerging topic of data streams
Low-Rank Matrices on Graphs: Generalized Recovery & Applications
Many real world datasets subsume a linear or non-linear low-rank structure in
a very low-dimensional space. Unfortunately, one often has very little or no
information about the geometry of the space, resulting in a highly
under-determined recovery problem. Under certain circumstances,
state-of-the-art algorithms provide an exact recovery for linear low-rank
structures but at the expense of highly inscalable algorithms which use nuclear
norm. However, the case of non-linear structures remains unresolved. We revisit
the problem of low-rank recovery from a totally different perspective,
involving graphs which encode pairwise similarity between the data samples and
features. Surprisingly, our analysis confirms that it is possible to recover
many approximate linear and non-linear low-rank structures with recovery
guarantees with a set of highly scalable and efficient algorithms. We call such
data matrices as \textit{Low-Rank matrices on graphs} and show that many real
world datasets satisfy this assumption approximately due to underlying
stationarity. Our detailed theoretical and experimental analysis unveils the
power of the simple, yet very novel recovery framework \textit{Fast Robust PCA
on Graphs
Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians
A key feature of ground states of gapped local 1D Hamiltonians is their
relatively low entanglement --- they are well approximated by matrix product
states (MPS) with bond dimension scaling polynomially in the length of the
chain, while general states require a bond dimension scaling exponentially. We
show that the bond dimension of these MPS approximations can be improved to a
constant, independent of the chain length, if we relax our notion of
approximation to be more local: for all length- segments of the chain, the
reduced density matrices of our approximations are -close to those of
the exact state. If the state is a ground state of a gapped local Hamiltonian,
the bond dimension of the approximation scales like ,
and at the expense of worse but still scaling of
the bond dimension, we give an alternate construction with the additional
features that it can be generated by a constant-depth quantum circuit with
nearest-neighbor gates, and that it applies generally for any state with
exponentially decaying correlations. For a completely general state, we give an
approximation with bond dimension , which is exponentially
worse, but still independent of . Then, we consider the prospect of
designing an algorithm to find a local approximation for ground states of
gapped local 1D Hamiltonians. When the Hamiltonian is translationally
invariant, we show that the ability to find -accurate local
approximations to the ground state in time implies the ability to
estimate the ground state energy to precision in time.Comment: 24 pages, 3 figures. v2: Theorem 1 extended to include construction
for general states; Lemma 7 & Theorem 2 slightly improved; figures added;
lemmas rearranged for clarity; typos fixed. v3: Reformatted & additional
references inserte
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