426 research outputs found
Tensor Network methods in many-body physics
Strongly correlated systems exhibit phenomena -- such as high-T_c superconductivity or the fractional quantum Hall effect -- that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime.
Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems.
It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems.
Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases -- symmetry-protected topological (SPT) order, -- and tensor networks provide the right language to classify these phases. In one dimension and for certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.Stark korrelierte Systeme zeigen Phänomene wie Hochtemperatursupraleitung oder den Quanten-Hall-Effekt, die mit klassischen und semiklassischen Methoden nicht erklärbar sind.
Da die Dimension des zugrundeliegenden Hilbertraums exponentiell mit der Größe des Systems wächst, versagen viele der traditionellen Ansätze für derartige Systeme. Es ist daher notwendig, neuartige numerische und analytische Methoden zu entwickeln, die die Physik in diesem Bereich erklären können.
Tensor-Netzwerkzustände können diese Schwierigkeiten zum Teil überwinden, indem sie eine Familie von Zuständen definieren, die nur von einer kleinen Anzahl von Parametern abhängen. Diese Zustände tragen auf zwei Arten zur Lösung des Problems bei: Erstens werden sie als Variationsansatz in numerischen Algorithmen verwendet. Zweitens bieten sie einen analytischen Zugang zu einer großen Klasse genau lösbarer Modelle, von denen angenommen wird, dass sie alle möglichen Materiephasen repräsentieren. In der vorliegenden Arbeit werden mathematische Eigenschaften dieser Zustände untersucht, wodurch das Verständnis dafür, wie und warum Tensor-Netzwerke für die Beschreibung von Quantensystemen geeignet sind, vertieft wird.
Zunächst widmen wir uns der Frage, inwiefern Tensornetzwerke Grundzustände lokaler Hamiltonians darstellen können.
Diese Frage ist von grundlegender Bedeutung, da Variationsalgorithmen, die auf Tensornetzwerken basieren, nur dann akkurate Ergebnisse liefern können, wenn der Grundzustand nicht allzu weit von der zugrundeliegenden variationellen Mannigfaltigkeit entfernt ist. Zwar kann prinzipiell jeder Quantenzustand als Tensornetzwerkstatus beschrieben werden. Jedoch ist die Anzahl der für die Beschreibung erforderlichen Parameter möglicherweise extrem groß. Dies ist bei eindimensionalen Systemen nicht der Fall: Nur wenige Parameter sind erforderlich, um eine gute Näherung ihrer Grundzustände zu erhalten. Aufgrund dieser theoretische Grundlage kann darauf vertraut werden, dass die Ergebnisse der tensornetzwerkbasierten Algorithmen akkurat sind. Die Situation in zwei Dimensionen ist komplizierter, aber es ist bekannt, dass Grundzustände lokaler Hamiltonians als Tensornetzwerke mit subexponentiell vielen Parametern ausgedrückt werden können. In der vorliegenden Arbeit verbessern wir diese bestehenden Grenzen und verstärken die Behauptung, dass Tensornetzwerke geeignet ist, Vielteilchensysteme zu beschreiben.
Eine weitere zentrale Frage ist, wie Symmetrien des Systems wie Translationsinvarianz, Zeitumkehrsymmetrie oder lokale Symmetrie in Tensornetzwerken reflektiert werden können. Das Verständnis dieser Symmetrien vereinfacht einerseits die Beschreibung der Systeme, in denen diese Symmetrien auftreten. Auf der anderen Seite führt das Vorhandensein von Symmetrien zum Entstehen neuer Phasen - sogenannter “symmetry protected topological phases” (SPT) -, und Tensornetzwerke liefern die richtige Beschreibung, um diese Phasen zu klassifizieren. In einer Dimension und für bestimmte Klassen von zweidimensionalen Tensornetzwerken (Zustände, die von sogenannten injektiven Tensoren erzeugt werden) ist es gut verstanden, wie Symmetrien des physikalischen System sich in ihrer Beschreibung als Tensornetzwerk widerspiegeln. Ein allgemeiner Rahmen muss jedoch noch entwickelt werden. In der vorliegenden Arbeit tragen wir auf zweierlei Weise zur Weiterentwicklung der Theorie bei. Wir untersuchen zunächst die Frage nach injektiven Tensoren und verallgemeinern den vorhandenen Beweis für jede Geometrie, einschließlich der in der AdS / CFT-Korrespondenz verwendeten hyperbolischen Geometrie. Zweitens führen wir eine Klasse von Tensornetzwerkzuständen ein, die bereits bekannte Beispiele für Zustände mit SPT-Ordnung enthalten. Wir zeigen, wie sich Symmetrien in diesen Zuständen widerspiegeln, wodurch das Verständnis der SPT-Ordnung in zwei Dimensionen vertieft wird
Algorithmic approaches for the single individual haplotyping problem
Since its introduction in 2001, the Single Individual Haplotyping problem has received an ever-increasing attention from the scientific community. In this paper we survey, in the form of an annotated bibliography, the developments in the study of the problem from its origin until our days
A Connectivity-Sensitive Approach to Consensus Dynamics
The paper resolves a long-standing open question in network dynamics. Averaging-based consensus has long been known to exhibit an exponential gap in relaxation time between the connected and disconnected cases, but a satisfactory explanation has remained elusive. We provide one by deriving nearly tight bounds on the s-energy of disconnected systems. This in turn allows us to relate the convergence rate of consensus dynamics to the number of connected components. We apply our results to opinion formation in social networks and provide a theoretical validation of the concept of an Overton window as an attracting manifold of "viable" opinions
Finding Similar Protein Structures Efficiently and Effectively
To assess the similarities and the differences among protein structures, a
variety of structure alignment algorithms and programs have been designed and
implemented. We introduce a low-resolution approach and a high-resolution
approach to evaluate the similarities among protein structures. Our results
show that both the low-resolution approach and the high-resolution approach
outperform state-of-the-art methods.
For the low-resolution approach, we eliminate false positives through the
comparison of both local similarity and remote similarity with little
compromise in speed. Two kinds of contact libraries (ContactLib) are introduced
to fingerprint protein structures effectively and efficiently. Each contact
group from the contact library consists of one local or two remote fragments
and is represented by a concise vector. These vectors are then indexed and used
to calculate a new combined hit-rate score to identify similar protein
structures effectively and efficiently.
We tested our ContactLibs on the high-quality protein structure subset of
SCOP30, which contains 3,297 protein structures. For each protein structure
of the subset, we retrieved its neighbor protein structures from the rest of
the subset. The best area under the ROC curve, archived by a ContactLib, is as
high as 0.960. This is a significant improvement over 0.747, the best
result achieved by the state-of-the-art method, FragBag.
For the high-resolution approach, our PROtein STructure Alignment method
(PROSTA) relies on and verifies the fact that the optimal protein structure
alignment always contains a small subset of aligned residue pairs, called a
seed, such that the rotation and translation (ROTRAN), which minimizes the RMSD
of the seed, yields both the optimal ROTRAN and the optimal alignment score.
Thus, ROTRANs minimizing the RMSDs of small subsets of residues are sampled,
and global alignments are calculated directly from the sampled ROTRANs.
Moreover, our method incorporates remote information and filters similar
ROTRANs (or alignments) by clustering, rather than by an exhaustive method, to
overcome the computational inefficiency.
Our high-resolution protein structure alignment method, when applied to
optimizing the TM-score and the GDT-TS score, produces a significantly better
result than state-of-the-art protein structure alignment methods.
Specifically, if the highest TM-score found by TM-align is lower than 0.6 and
the highest TM-score found by one of the tested methods is higher than 0.5,
our alignment method tends to discover better protein structure alignments with
(up to 0.21) higher TM-scores. In such cases, TM-align fails to find TM-scores
higher than 0.5 with a probability of 42%; however, our alignment method
fails the same task with a probability of only 2%.
In addition, existing protein structure alignment scoring functions focus on
atom coordinate similarity alone and simply ignore other important
similarities, such as sequence similarity. Our scoring function has the
capacity for incorporating multiple similarities into the scoring function. Our
result shows that sequence similarity aids in finding high quality protein
structure alignments that are more consistent with HOMSTRAD alignments, which
are protein structure alignments examined by human experts. When atom
coordinate similarity itself fails to find alignments with any consistency to
HOMSTRAD alignments, our scoring function remains capable of finding alignments
highly similar to, or even identical to, HOMSTRAD alignments
Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
The theory of entanglement provides a fundamentally new language for
describing interactions and correlations in many body systems. Its vocabulary
consists of qubits and entangled pairs, and the syntax is provided by tensor
networks. We review how matrix product states and projected entangled pair
states describe many-body wavefunctions in terms of local tensors. These
tensors express how the entanglement is routed, act as a novel type of
non-local order parameter, and we describe how their symmetries are reflections
of the global entanglement patterns in the full system. We will discuss how
tensor networks enable the construction of real-space renormalization group
flows and fixed points, and examine the entanglement structure of states
exhibiting topological quantum order. Finally, we provide a summary of the
mathematical results of matrix product states and projected entangled pair
states, highlighting the fundamental theorem of matrix product vectors and its
applications.Comment: Review article. 72 page
Data driven approach to detection of quantum phase transitions
Phase transitions are fundamental phenomena in (quantum) many-body systems. They are associated with changes in the macroscopic physical properties of the system in response to the alteration in the conditions controlled by one or more parameters, like temperature or coupling constants. Quantum phase transitions are particularly intriguing as they reveal new insights into the fundamental nature of matter and the laws of physics. The study of phase transitions in such systems is crucial in aiding our understanding of how materials behave in extreme conditions, which are difficult to replicate in laboratory, and also the behaviour of exotic states of matter with unique and potentially useful properties like superconductors and superfluids. Moreover, this understanding has other practical applications and can lead to the development of new materials with specific properties or more efficient technologies, such as quantum computers. Hence, detecting the transition point from one phase of matter to another and constructing the corresponding phase diagram is of great importance for examining many-body systems and predicting their response to external perturbations.
Traditionally, phase transitions have been identified either through analytical methods like mean field theory or numerical simulations. The pinpointing of the critical value normally involves the measure of specific quantities such as local observables, correlation functions, energy gaps, etc. reflecting the changes in the physics through the transition. However, the latter approach requires prior knowledge of the system to calculate the order parameter of the transition, which is uniquely associated to its universality class. Recently, another method has gained more and more attention in the physics community. By using raw and very general representative data of the system, one can resort to machine learning techniques to distinguish among patterns within the data belonging to different phases. The relevance of these techniques is rooted in the ability of a properly trained machine to efficiently process complex data for the sake of pursuing classification tasks, pattern recognition, generating brand new data and even developing decision processes.
The aim of this thesis is to explore phase transitions from this new and promising data-centric perspective. On the one hand, our work is focused on the development of new machine learning architectures using state-of-the-art and interpretable models. On the other hand, we are interested in the study of the various possible data which can be fed to the artificial intelligence model for the mapping of a quantum many-body system phase diagram. Our analysis is supported by numerical examples obtained via matrix-product-states (MPS) simulations for several one-dimensional zero-temperature systems on a lattice such as the XXZ model, the Extended Bose-Hubbard model (EBH) and the two-species Bose Hubbard model (BH2S)
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