1,921 research outputs found
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
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-dimension 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.Comment: 115 pages, 56 figure
One-dimensional many-body entangled open quantum systems with tensor network methods
We present a collection of methods to simulate entangled dynamics of open
quantum systems governed by the Lindblad equation with tensor network methods.
Tensor network methods using matrix product states have been proven very useful
to simulate many-body quantum systems and have driven many innovations in
research. Since the matrix product state design is tailored for closed
one-dimensional systems governed by the Schr\"odinger equation, the next step
for many-body quantum dynamics is the simulation of open quantum systems. We
review the three dominant approaches to the simulation of open quantum systems
via the Lindblad master equation: quantum trajectories, matrix product density
operators, and locally purified tensor networks. Selected examples guide
possible applications of the methods and serve moreover as a benchmark between
the techniques. These examples include the finite temperature states of the
transverse quantum Ising model, the dynamics of an exciton traveling under the
influence of spontaneous emission and dephasing, and a double-well potential
simulated with the Bose-Hubbard model including dephasing. We analyze which
approach is favorable leading to the conclusion that a complete set of all
three methods is most beneficial, push- ing the limits of different scenarios.
The convergence studies using analytical results for macroscopic variables and
exact diagonalization methods as comparison, show, for example, that matrix
product density operators are favorable for the exciton problem in our study.
All three methods access the same library, i.e., the software package Open
Source Matrix Product States, allowing us to have a meaningful comparison
between the approaches based on the selected examples. For example, tensor
operations are accessed from the same subroutines and with the same
optimization eliminating one possible bias in a comparison of such numerical
methods.Comment: 24 pages, 8 figures. Small extension of time evolution section and
moving quantum simulators to introduction in comparison to v
Tensor decompositions for learning latent variable models
This work considers a computationally and statistically efficient parameter
estimation method for a wide class of latent variable models---including
Gaussian mixture models, hidden Markov models, and latent Dirichlet
allocation---which exploits a certain tensor structure in their low-order
observable moments (typically, of second- and third-order). Specifically,
parameter estimation is reduced to the problem of extracting a certain
(orthogonal) decomposition of a symmetric tensor derived from the moments; this
decomposition can be viewed as a natural generalization of the singular value
decomposition for matrices. Although tensor decompositions are generally
intractable to compute, the decomposition of these specially structured tensors
can be efficiently obtained by a variety of approaches, including power
iterations and maximization approaches (similar to the case of matrices). A
detailed analysis of a robust tensor power method is provided, establishing an
analogue of Wedin's perturbation theorem for the singular vectors of matrices.
This implies a robust and computationally tractable estimation approach for
several popular latent variable models
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