scholarly journals Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 556
Author(s):  
Giovanni Cataldi ◽  
Ashkan Abedi ◽  
Giuseppe Magnifico ◽  
Simone Notarnicola ◽  
Nicola Dalla Pozza ◽  
...  
Keyword(s):  
Transverse Field ◽  
Matrix Product ◽  
Hilbert Curve ◽  
Many Body ◽  
Network Efficiency ◽  
Lattice Systems ◽  
Tensor Network ◽  
Quantum Ising Model ◽  
1D Chain ◽  
Numerical Precision

We present a novel mapping for studying 2D many-body quantum systems by solving an effective, one-dimensional long-range model in place of the original two-dimensional short-range one. In particular, we address the problem of choosing an efficient mapping from the 2D lattice to a 1D chain that optimally preserves the locality of interactions within the TN structure. By using Matrix Product States (MPS) and Tree Tensor Network (TTN) algorithms, we compute the ground state of the 2D quantum Ising model in transverse field with lattice size up to 64×64, comparing the results obtained from different mappings based on two space-filling curves, the snake curve and the Hilbert curve. We show that the locality-preserving properties of the Hilbert curve leads to a clear improvement of numerical precision, especially for large sizes, and turns out to provide the best performances for the simulation of 2D lattice systems via 1D TN structures.

scholarly journals Computable Rényi mutual information: Area laws and correlations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 541
Author(s):  
Samuel O. Scalet ◽  
Álvaro M. Alhambra ◽  
Georgios Styliaris ◽  
J. Ignacio Cirac
Keyword(s):  
Mutual Information ◽  
Correlation Functions ◽  
Quantum Correlations ◽  
Matrix Product ◽  
Many Body ◽  
Von Neumann ◽  
Tensor Network ◽  
Network States ◽  
Tensor Network States ◽  
Area Law

The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states like matrix product operators while preserving all desirable properties of a measure of correlations. In particular, we show that they obey a thermal area law in great generality, and that they upper bound all correlation functions. We also investigate their behavior on certain tensor network states and on classical thermal distributions.

scholarly journals Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 528
Author(s):  
Alberto Biella ◽  
Marco Schiró
Keyword(s):  
Continuous Monitoring ◽  
Transverse Field ◽  
Body System ◽  
Ising Chain ◽  
Quantum Zeno Effect ◽  
Many Body ◽  
Unitary Evolution ◽  
Zeno Effect ◽  
Quantum Ising Model ◽  
Sharp Phase Transition

It is well known that by repeatedly measuring a quantum system it is possible to completely freeze its dynamics into a well defined state, a signature of the quantum Zeno effect. Here we show that for a many-body system evolving under competing unitary evolution and variable-strength measurements the onset of the Zeno effect takes the form of a sharp phase transition. Using the Quantum Ising chain with continuous monitoring of the transverse magnetization as paradigmatic example we show that for weak measurements the entanglement produced by the unitary dynamics remains protected, and actually enhanced by the monitoring, while only above a certain threshold the system is sharply brought into an uncorrelated Zeno state. We show that this transition is invisible to the average dynamics, but encoded in the rare fluctuations of the stochastic measurement process, which we show to be perfectly captured by a non-Hermitian Hamiltonian which takes the form of a Quantum Ising model in an imaginary valued transverse field. We provide analytical results based on the fermionization of the non-Hermitian Hamiltonian in supports of our exact numerical calculations.

scholarly journals Studying dynamics in two-dimensional quantum lattices using tree tensor network states

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Benedikt Kloss ◽  
David Reichman ◽  
Yevgeny Bar Lev
Keyword(s):  
Exact Algorithm ◽  
Matrix Product ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Lattice Systems ◽  
Matrix Product State ◽  
Tensor Network ◽  
Network States ◽  
Quantum Lattices ◽  
Tensor Network States

We analyze and discuss convergence properties of a numerically exact algorithm tailored to study the dynamics of interacting two-dimensional lattice systems. The method is based on the application of the time-dependent variational principle in a manifold of binary and quaternary Tree Tensor Network States. The approach is found to be competitive with existing matrix product state approaches. We discuss issues related to the convergence of the method, which could be relevant to a broader set of numerical techniques used for the study of two-dimensional systems.

scholarly journals Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

2018 ◽  
Author(s):  
Johannes Hauschild ◽  
Frank Pollmann
Keyword(s):  
Matrix Product ◽  
Many Body ◽  
Density Matrix Renormalization ◽  
Tensor Networks ◽  
Condensed Matter Theory ◽  
Tensor Network ◽  
Matter Theory ◽  
Particle Number Conservation ◽  
Many Body Systems ◽  
The One

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

scholarly journals The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

2019 ◽  
Author(s):  
Pietro Silvi ◽  
Ferdinand Tschirsich ◽  
Matthias Gerster ◽  
Johannes Jünemann ◽  
Daniel Jaschke ◽  
...  
Keyword(s):  
Data Structures ◽  
Finite Size ◽  
Many Body ◽  
Lattice Systems ◽  
Arbitrary Boundary ◽  
The Core ◽  
Simulation Techniques ◽  
Tensor Networks ◽  
Tensor Network ◽  
Network Methods

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimensional physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer’s companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

scholarly journals Quantum Ising model in a period-2 modulated transverse field

2019 ◽  
Vol 100 (2) ◽  
Author(s):  
Adalberto D. Varizi ◽  
Raphael C. Drumond
Keyword(s):  
Ising Model ◽  
Transverse Field ◽  
Period 2 ◽  
Quantum Ising Model

scholarly journals Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 984
Author(s):  
Regina Finsterhölzl ◽  
Manuel Katzer ◽  
Andreas Knorr ◽  
Alexander Carmele
Keyword(s):  
Time Evolution ◽  
Nearest Neighbor ◽  
Matrix Product ◽  
Body System ◽  
Many Body ◽  
Initial State ◽  
Matrix Product States ◽  
Open Quantum ◽  
Many Body Systems ◽  
The Many

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.

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