scholarly journals Automated construction of $U(1)$-invariant matrix-product operators from graph representations

2017 ◽  
Vol 3 (5) ◽  
Author(s):  
Sebastian Paeckel ◽  
Thomas Köhler ◽  
Salvatore R. Manmana
Keyword(s):  
Building Blocks ◽  
Matrix Product ◽  
Heisenberg Chain ◽  
Graph Representations ◽  
Quantum Numbers ◽  
Construction Scheme ◽  
Product Operator ◽  
Local Quantum ◽  
Local Operators ◽  
Algorithmic Construction

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary U(1)U(1)-invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a U(1)U(1)-invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of U(1)U(1)-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a S=1S=1 Heisenberg chain.

IBCube

2018 ◽  
Vol 15 (1) ◽  
pp. 27-46
Author(s):  
Qiong Hu ◽  
Hanhua Chen ◽  
Hai Jin ◽  
Chen Tian ◽  
Aobing Sun ◽  
...  
Keyword(s):  
Low Cost ◽  
Packet Delay ◽  
Building Blocks ◽  
Network Interface ◽  
Allocation Scheme ◽  
The Past ◽  
Datacenter Networks ◽  
Construction Scheme ◽  
Economical Design ◽  
Simulation Results

Datacenter networks have attracted a lot of research interest in the past few years. BCube is proved to be a promising scheme due to its low cost. By using a recursive construction scheme, BCube can exponentially scale a datacenter. Industry experiences, however, articulate the importance of incremental expansion of datacenter. In this article, the authors show that BCube's expanding scheme suffers low utilization of switch ports. They propose IBCube, a novel economical design for incrementally building datacenter networks. The insight is that: by letting the number of switches in each BCube layer equal the number of the building blocks, the authors can enable the switch ports to be fully utilized to support the total number of network interface cards of the deployed servers in the datacenters. Accordingly, their IBCube designs a novel automatic port allocation scheme. Simulation results show that the IBCube design reduces the budget for the datacenter networks by 94% as well as improves the packet delay and throughput by 10.3% and 11.5%, respectively, compared to the previous partial BCube design.

Applying Rigidity Theory Methods for Topological Decomposition and Synthesis of Gear Train Systems

2012 ◽  
Author(s):  
Maria Terushkin ◽  
Offer Shai
Keyword(s):  
Synthesis Method ◽  
Building Blocks ◽  
The Body ◽  
Computational Method ◽  
Gear Train ◽  
Graph Representation ◽  
Rigidity Theory ◽  
Graph Representations ◽  
Machine Theory ◽  
Gear Trains

This paper introduces a novel way to augment the knowledge and methods of rigidity theory to the topological decomposition and synthesis of gear train systems. A graph of gear trains, widely reported in the literature of machine theory, is treated as a graph representation from rigidity theory—the Body-Bar graph. Once we have this Body-Bar graph, methods and theorems from rigidity theory can be employed for analysis and synthesis. In this paper we employ the pebble-game algorithm, a computational method which allows determination of the topological mobility of mechanisms and the decomposition of gear trains into basic building blocks—Body-Bar Assur Graphs. Once we gain the ability to decompose any gear train into standalone components (Body-Bar Assur Graphs), this paper suggests inverting the process and applying the same method for synthesis. Relying on rigidity theory operations (Body-Bar extension, in this case), it is possible to construct all of the Body-Bar Assur Graphs, meaning the building blocks of gear trains. Once we have these building blocks at hand, it is possible to recombine them in various ways, providing us with a topological synthesis method for constructing gear trains. This paper also introduces a transformation between the Body-Bar graph and other graph representations used in mechanisms, thus leaving room for the application of the proposed synthesis and decomposition method directly to known graph representations already used in machine theory.

scholarly journals Anyons and matrix product operator algebras

2017 ◽  
Vol 378 ◽  
pp. 183-233 ◽  
Author(s):  
N. Bultinck ◽  
M. Mariën ◽  
D.J. Williamson ◽  
M.B. Şahinoğlu ◽  
J. Haegeman ◽  
...  
Keyword(s):  
Operator Algebras ◽  
Matrix Product ◽  
Product Operator

A SYSTEMATIC APPROACH TO GENERATING N-SCROLL ATTRACTORS

2002 ◽  
Vol 12 (12) ◽  
pp. 2907-2915 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM-FUNG MAN ◽  
GUANRONG CHEN
Keyword(s):  
Systematic Approach ◽  
Building Blocks ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Nonlinear Resistor ◽  
Construction Scheme ◽  
Break Points ◽  
Set Up

A new circuitry design based on Chua's circuit for generating n-scroll attractors (n = 1, 2, 3, …) is proposed. In this design, the nonlinear resistor in Chua's circuit is constructed via a systematical procedure using basic building blocks. With the proposed construction scheme, the slopes and break points of the v–i characteristic of the circuit can be tuned independently, and chaotic attractors with an even or an odd number of scrolls can be easily generated. Distinct attractors with n-scrolls (n = 5, 6, 7, 8, 9, 10) obtained with this simple experimental set-up are demonstrated.

scholarly journals GraphMineSuite

2021 ◽  
Vol 14 (11) ◽  
pp. 1922-1935
Author(s):  
Maciej Besta ◽  
Zur Vonarburg-Shmaria ◽  
Yannick Schaffner ◽  
Leonardo Schwarz ◽  
Grzegorz Kwasniewski ◽  
...  
Keyword(s):  
Graph Mining ◽  
High Performance ◽  
Maximal Clique ◽  
Building Blocks ◽  
Extensive Literature ◽  
Performance Bounds ◽  
Graph Representations ◽  
Fine Grained ◽  
Performance Metric ◽  
Mining Algorithms

We propose GraphMineSuite (GMS): the first benchmarking suite for graph mining that facilitates evaluating and constructing high-performance graph mining algorithms. First, GMS comes with a benchmark specification based on extensive literature review, prescribing representative problems, algorithms, and datasets. Second, GMS offers a carefully designed software platform for seamless testing of different fine-grained elements of graph mining algorithms, such as graph representations or algorithm subroutines. The platform includes parallel implementations of more than 40 considered baselines, and it facilitates developing complex and fast mining algorithms. High modularity is possible by harnessing set algebra operations such as set intersection and difference, which enables breaking complex graph mining algorithms into simple building blocks that can be separately experimented with. GMS is supported with a broad concurrency analysis for portability in performance insights, and a novel performance metric to assess the throughput of graph mining algorithms, enabling more insightful evaluation. As use cases, we harness GMS to rapidly redesign and accelerate state-of-the-art baselines of core graph mining problems: degeneracy reordering (by >2X), maximal clique listing (by >9×), k -clique listing (by up to 1.1×), and subgraph isomorphism (by 2.5×), also obtaining better theoretical performance bounds.

scholarly journals Asymptotic Performance of Port-Based Teleportation

2020 ◽  
Author(s):  
Matthias Christandl ◽  
Felix Leditzky ◽  
Christian Majenz ◽  
Graeme Smith ◽  
Florian Speelman ◽  
...  
Keyword(s):  
Matrix Theory ◽  
Building Blocks ◽  
Convergence Result ◽  
Dirichlet Eigenvalue ◽  
Shannon Theory ◽  
Optimal Protocol ◽  
Local Quantum ◽  
Non Local ◽  
Fixed Input ◽  
Quantum Shannon Theory

AbstractQuantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur–Weyl distribution by Johansson, which might be of independent interest.

Scaling and efficient classical simulation of the quantum Fourier transform

2017 ◽  
Vol 17 (1&2) ◽  
pp. 1-14
Author(s):  
Kieran J. Woolfe ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Fourier Transform ◽  
Matrix Product ◽  
Product State ◽  
Quantum Fourier Transform ◽  
Numerical Evidence ◽  
Matrix Product State ◽  
Product Operator ◽  
Classical Simulation ◽  
Schmidt Rank

We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with n qubits of maximum Schmidt rank χ can be simulated in O(n (log(n))2 χ 2 ) time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.

scholarly journals Matrix product operator representations

2010 ◽  
Vol 12 (2) ◽  
pp. 025012 ◽  
Author(s):  
B Pirvu ◽  
V Murg ◽  
J I Cirac ◽  
F Verstraete
Keyword(s):  
Matrix Product ◽  
Product Operator ◽  
Operator Representations
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