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 Machine Learning Tensor Network States

2021 ◽  
Vol 8 ◽  
Author(s):  
Andrey Kardashin ◽  
Alexey Uvarov ◽  
Jacob Biamonte
Keyword(s):  
Machine Learning ◽  
Quantum Algorithm ◽  
Unitary Matrix ◽  
Many Body ◽  
Network Algorithms ◽  
Tensor Networks ◽  
Tensor Network ◽  
Network Methods ◽  
Network States ◽  
Tensor Network States

Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.

scholarly journals Hyper-optimized tensor network contraction

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 410
Author(s):  
Johnnie Gray ◽  
Stefanos Kourtis
Keyword(s):  
Computation Time ◽  
Quantum Circuit ◽  
Optimization Approach ◽  
Many Body ◽  
Tensor Networks ◽  
Randomized Protocols ◽  
Classical Simulation ◽  
Tensor Network ◽  
Speed Up ◽  
Many Body Systems

Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to tensor networks with irregular geometries. Finding the best possible contraction path for such networks is a central problem, with an exponential effect on computation time and memory footprint. In this work, we implement new randomized protocols that find very high quality contraction paths for arbitrary and large tensor networks. We test our methods on a variety of benchmarks, including the random quantum circuit instances recently implemented on Google quantum chips. We find that the paths obtained can be very close to optimal, and often many orders or magnitude better than the most established approaches. As different underlying geometries suit different methods, we also introduce a hyper-optimization approach, where both the method applied and its algorithmic parameters are tuned during the path finding. The increase in quality of contraction schemes found has significant practical implications for the simulation of quantum many-body systems and particularly for the benchmarking of new quantum chips. Concretely, we estimate a speed-up of over 10,000× compared to the original expectation for the classical simulation of the Sycamore `supremacy' circuits.

scholarly journals Automatic contraction of unstructured tensor networks

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Adam Jermyn
Keyword(s):  
Partition Function ◽  
Statistical Physics ◽  
Discrete Models ◽  
Correlation Structure ◽  
Central Problem ◽  
Partition Functions ◽  
Lattice Systems ◽  
Tensor Networks ◽  
Tensor Network

The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local.

scholarly journals Loop-TNR analysis of CP(1) model with theta term

2018 ◽  
Vol 175 ◽  
pp. 11015
Author(s):  
Hikaru Kawauchi ◽  
Shinji Takeda
Keyword(s):  
Renormalization Group ◽  
Phase Structure ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Tensor Networks ◽  
Tensor Network ◽  
Network Methods ◽  
Renormalization Method ◽  
Future Work

The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.

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 Mutual Information Scaling for Tensor Network Machine Learning

2021 ◽  
Author(s):  
Ian Convy ◽  
William Huggins ◽  
Haoran Liao ◽  
K Birgitta Whaley
Keyword(s):  
Machine Learning ◽  
Mutual Information ◽  
Many Body ◽  
Learning Tasks ◽  
Tensor Networks ◽  
Tensor Network ◽  
Logistic Regression Algorithm ◽  
Entanglement Structure ◽  
Insight Into ◽  
Specific Learning

Abstract Tensor networks have emerged as promising tools for machine learning, inspired by their widespread use as variational ansatze in quantum many-body physics. It is well known that the success of a given tensor network ansatz depends in part on how well it can reproduce the underlying entanglement structure of the target state, with different network designs favoring different scaling patterns. We demonstrate here how a related correlation analysis can be applied to tensor network machine learning, and explore whether classical data possess correlation scaling patterns similar to those found in quantum states which might indicate the best network to use for a given dataset. We utilize mutual information as measure of correlations in classical data, and show that it can serve as a lower-bound on the entanglement needed for a probabilistic tensor network classifier. We then develop a logistic regression algorithm to estimate the mutual information between bipartitions of data features, and verify its accuracy on a set of Gaussian distributions designed to mimic different correlation patterns. Using this algorithm, we characterize the scaling patterns in the MNIST and Tiny Images datasets, and find clear evidence of boundary-law scaling in the latter. This quantum-inspired classical analysis offers insight into the design of tensor networks which are best suited for specific learning tasks.

scholarly journals Automatic differentiation applied to excitations with projected entangled pair states

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Boris Ponsioen ◽  
Fakher Assaad ◽  
Philippe Corboz
Keyword(s):  
Automatic Differentiation ◽  
Optimization Methods ◽  
Two Dimensions ◽  
Strongly Correlated ◽  
Many Body ◽  
Tensor Networks ◽  
Tensor Network ◽  
New Ideas ◽  
Half Filling ◽  
Many Body Systems

The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.

scholarly journals Size consistency of tensor network methods for quantum many-body systems

2013 ◽  
Vol 88 (12) ◽  
Author(s):  
Zhen Wang ◽  
Yongjian Han ◽  
Guang-Can Guo ◽  
Lixin He
Keyword(s):  
Many Body ◽  
Size Consistency ◽  
Tensor Network ◽  
Network Methods ◽  
Many Body Systems

scholarly journals The seniority quantum number in Tensor Network States

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Klaas Gunst ◽  
Dimitri Van Neck ◽  
Peter Andreas Limacher ◽  
Stijn De Baerdemacker
Keyword(s):  
Quantum Number ◽  
Body Wave ◽  
Many Body ◽  
Dynamical Correlation ◽  
Molecular Systems ◽  
Tensor Network ◽  
Network Methods ◽  
Unpaired Electrons ◽  
Network States ◽  
Tensor Network States

We employ tensor network methods for the study of the seniority quantum number – defined as the number of unpaired electrons in a many-body wave function – in molecular systems. Seniority-zero methods recently emerged as promising candidates to treat strong static correlations in molecular systems, but are prone to deficiencies related to dynamical correlation and dispersion. We systematically resolve these deficiencies by increasing the allowed seniority number using tensor network methods. In particular, we investigate the number of unpaired electrons needed to correctly describe the binding of the neon and nitrogen dimer and the \mathbf{D_{6h}}D6h symmetry of benzene.

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