scholarly journals Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 328
Author(s):  
A. Nietner ◽  
B. Vanhecke ◽  
F. Verstraete ◽  
J. Eisert ◽  
L. Vanderstraeten
Keyword(s):  
Variational Principles ◽  
Valence Bond ◽  
Computational Effort ◽  
Unit Cell ◽  
Square Lattice ◽  
Partition Functions ◽  
Matrix Product ◽  
Two Dimensional ◽  
Matrix Product State ◽  
Tensor Networks

Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.

Irreducible first Brillouin zone for two-dimensional magnonic crystals with asymmetric complex lattices

2021 ◽  
pp. 2150170
Author(s):  
Hui Yang ◽  
Guo-Hong Yun ◽  
Yong-Jun Cao
Keyword(s):  
Brillouin Zone ◽  
Unit Cell ◽  
Square Lattice ◽  
Two Dimensional ◽  
Area Formula ◽  
Irreducible Part ◽  
Full Band ◽  
Magnonic Crystals ◽  
Magnonic Crystal ◽  
Complex Basis

Two-dimensional (2D) magnonic crystal (MC) with an asymmetric complex basis is proposed in this paper, and its band structures are calculated in the whole area of the first Brillouin zone (BZ). This kind of MCs is composed of two different atoms in the unit cell, and the symmetry of the unit cell is broken due to changes in the position of the second atom, so the irreducible part of the BZ is no longer the small area [Formula: see text] for square lattice, and it must be expanded to the whole first BZ. Only by investigating the whole first BZ, can we get the true full band-gap for this kind of MCs.

Dispersion characteristics of periodic structural systems using higher order beam element dynamics

2019 ◽  
Vol 25 (2) ◽  
pp. 457-474
Author(s):  
M Ayad ◽  
N Karathanasopoulos ◽  
H Reda ◽  
JF Ganghoffer ◽  
H Lakiss
Keyword(s):  
Dynamic Equilibrium ◽  
Dimensional Space ◽  
Higher Order ◽  
Unit Cell ◽  
Square Lattice ◽  
Structural Systems ◽  
Discrete Dynamics ◽  
Two Dimensional ◽  
Dispersion Characteristics ◽  
Order Expansion

In the current work, we elaborate upon a beam mechanics-based discrete dynamics approach for the computation of the dispersion characteristics of periodic structures. Within that scope, we compute the higher order asymptotic expansion of the forces and moments developed within beam structural elements upon dynamic loads. Thereafter, we employ the obtained results to compute the dispersion characteristics of one- and two-dimensional periodic media. In the one-dimensional space, we demonstrate that single unit-cell equilibrium can provide the fundamental low-frequency band diagram structure, which can be approximated by non-dispersive Cauchy media formulations. However, we show that the discrete dynamics method can access the higher frequency modes by considering multiple unit-cell systems for the dynamic equilibrium, frequency ranges that cannot be accessed by simplified formulations. We extend the analysis into two-dimensional space computing with the dispersion attributes of square lattice structures. Thereupon, we demonstrate that the discrete dynamics dispersion results compare well with that obtained using Bloch theorem computations. We show that a high-order expansion of the inner element forces and moments of the structures is required for the higher wave propagation modes to be accurately represented, in contrast to the shear and the longitudinal mode, which can be captured using a lower, fourth-order expansion of its inner dynamic forces and moments. The provided results can serve as a reference analysis for the computation of the dispersion characteristics of periodic structural systems with the use of discrete element dynamics.

A Two-Dimensional Coordination Polymer Formed from Cobalt(II) and an Extended Dipyridyl Ligand

2020 ◽  
Vol 73 (6) ◽  
pp. 547 ◽  
Author(s):  
Hydar A. AL-Fayaad ◽  
Rashid G. Siddique ◽  
Kasun S. Athukorala Arachchige ◽  
Jack K. Clegg
Keyword(s):  
Coordination Polymer ◽  
Cell Volume ◽  
Crystal Structures ◽  
Unit Cell Volume ◽  
Suzuki Coupling ◽  
Unit Cell ◽  
Square Lattice ◽  
Two Dimensional ◽  
Cobalt Ions ◽  
One Dimensional

The synthesis of the extended dipyridyl ligand 4,4′-(2,5-dimethyl-1,4-phenylene)dipyridine (L) in an improved yield via the palladium catalysed Suzuki coupling of 4-(4,4,5,5-tetramethyl-1,3,2-dioxaborolan-2-yl)pyridine (1) and 1,4-dibromo-2,5-dimethylbenzene (2) is reported along with its use to form a two-dimensional coordination polymer [Co2L2(OAc)4(H2O)2]n. The coordination polymer consists of one-dimensional chains of octahedral cobalt ions bridged by acetate ligands which are connected to form two dimensional sheets with square lattice (sql) topology via the dipyridyl ligands (L). The structure contains small voids totalling ~6.6% of the unit cell volume. The crystal structures 1, L, L·2H2O, and L·2HNO3 are also reported.

Entangled spin chain

2017 ◽  
Vol 29 (10) ◽  
pp. 1750031 ◽  
Author(s):  
Olof Salberger ◽  
Vladimir Korepin
Keyword(s):  
Entanglement Entropy ◽  
Square Lattice ◽  
Symmetric Model ◽  
Matrix Product ◽  
Reversible Computing ◽  
Half Integer ◽  
Matrix Product State ◽  
Fredkin Gate ◽  
Leading Term ◽  
Higher Spins

We introduce a new model of interacting spin 1/2. It describes interactions of three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin gates. The Fredkin gate (also known as the controlled swap gate) is a computational circuit suitable for reversible computing. Our construction generalizes the model presented by Peter Shor and Ramis Movassagh to half-integer spins. Our model can be solved by means of Catalan combinatorics in the form of random walks on the upper half plane of a square lattice (Dyck walks). Each Dyck path can be mapped on a wave function of spins. The ground state is an equally weighted superposition of Dyck walks (instead of Motzkin walks). We can also express it as a matrix product state. We further construct a model of interacting spins 3/2 and greater half-integer spins. The models with higher spins require coloring of Dyck walks. We construct a [Formula: see text] symmetric model (where [Formula: see text] is the number of colors). The leading term of the entanglement entropy is then proportional to the square root of the length of the lattice (like in the Shor–Movassagh model). The gap closes as a high power of the length of the lattice [5, 11].

scholarly journals Parallel quantum simulation of large systems on small NISQ computers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
F. Barratt ◽  
James Dborin ◽  
Matthias Bal ◽  
Vid Stojevic ◽  
Frank Pollmann ◽  
...  
Keyword(s):  
Finite Depth ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum Spin ◽  
Matrix Product ◽  
One Dimensional ◽  
Matrix Product State ◽  
Tensor Networks ◽  
Large Systems ◽  
Invariant Quantum

AbstractTensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given.

scholarly journals Abelian and non-abelian symmetries in infinite projected entangled pair states

2018 ◽  
Vol 5 (5) ◽  
Author(s):  
Claudius Hubig
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Ground State ◽  
Heisenberg Model ◽  
Computational Effort ◽  
Square Lattice ◽  
Two Dimensional ◽  
Computational Speed ◽  
Speed Up ◽  
Ground State Energies

We explore in detail the implementation of arbitrary abelian and non-abelian symmetries in the setting of infinite projected entangled pair states on the two-dimensional square lattice. We observe a large computational speed-up; easily allowing bond dimensions D=10D=10 in the square lattice Heisenberg model at computational effort comparable to calculations at D=6D=6 without symmetries. We also find that implementing an unbroken symmetry does not negatively affect the representative power of the state and leads to identical or improved ground-state energies. Finally, we point out how to use symmetry implementations to detect spontaneous symmetry breaking.

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 Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Jack Y. Araz ◽  
Michael Spannowsky
Keyword(s):  
Top Quark ◽  
Entanglement Entropy ◽  
Feature Space ◽  
Machine Learning Techniques ◽  
Stochastic Gradient Descent ◽  
Matrix Product ◽  
Density Matrix Renormalization ◽  
Matrix Product State ◽  
Tensor Networks ◽  
Many Body Systems

Abstract Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine learning techniques, thereby facilitating an improved interpretability of neural networks. This study presents the discrimination of top quark signal over QCD background processes using a Matrix Product State classifier. We show that entanglement entropy can be used to interpret what a network learns, which can be used to reduce the complexity of the network and feature space without loss of generality or performance. For the optimisation of the network, we compare the Density Matrix Renormalization Group (DMRG) algorithm to stochastic gradient descent (SGD) and propose a joined training algorithm to harness the explainability of DMRG with the efficiency of SGD.

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