11,106 research outputs found
Gauging tensor networks with belief propagation
Effectively compressing and optimizing tensor networks requires reliable
methods for fixing the latent degrees of freedom of the tensors, known as the
gauge. Here we introduce a new algorithm for gauging tensor networks using
belief propagation, a method that was originally formulated for performing
statistical inference on graphical models and has recently found applications
in tensor network algorithms. We show that this method is closely related to
known tensor network gauging methods. It has the practical advantage, however,
that existing belief propagation implementations can be repurposed for tensor
network gauging, and that belief propagation is a very simple algorithm based
on just tensor contractions so it can be easier to implement, optimize, and
generalize. We present numerical evidence and scaling arguments that this
algorithm is faster than existing gauging algorithms, demonstrating its usage
on structured, unstructured, and infinite tensor networks. Additionally, we
apply this method to improve the accuracy of the widely used simple update gate
evolution algorithm.Comment: 47 Pages. 11 Figure
Duality of Graphical Models and Tensor Networks
In this article we show the duality between tensor networks and undirected
graphical models with discrete variables. We study tensor networks on
hypergraphs, which we call tensor hypernetworks. We show that the tensor
hypernetwork on a hypergraph exactly corresponds to the graphical model given
by the dual hypergraph. We translate various notions under duality. For
example, marginalization in a graphical model is dual to contraction in the
tensor network. Algorithms also translate under duality. We show that belief
propagation corresponds to a known algorithm for tensor network contraction.
This article is a reminder that the research areas of graphical models and
tensor networks can benefit from interaction
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Kernel Belief Propagation
We propose a nonparametric generalization of belief propagation, Kernel
Belief Propagation (KBP), for pairwise Markov random fields. Messages are
represented as functions in a reproducing kernel Hilbert space (RKHS), and
message updates are simple linear operations in the RKHS. KBP makes none of the
assumptions commonly required in classical BP algorithms: the variables need
not arise from a finite domain or a Gaussian distribution, nor must their
relations take any particular parametric form. Rather, the relations between
variables are represented implicitly, and are learned nonparametrically from
training data. KBP has the advantage that it may be used on any domain where
kernels are defined (Rd, strings, groups), even where explicit parametric
models are not known, or closed form expressions for the BP updates do not
exist. The computational cost of message updates in KBP is polynomial in the
training data size. We also propose a constant time approximate message update
procedure by representing messages using a small number of basis functions. In
experiments, we apply KBP to image denoising, depth prediction from still
images, and protein configuration prediction: KBP is faster than competing
classical and nonparametric approaches (by orders of magnitude, in some cases),
while providing significantly more accurate results
Efficient tensor network simulation of IBM's Eagle kicked Ising experiment
We report an accurate and efficient classical simulation of a kicked Ising
quantum system on the heavy-hexagon lattice. A simulation of this system was
recently performed on a 127 qubit quantum processor using noise mitigation
techniques to enhance accuracy (Nature volume 618, p.500-505 (2023)). Here we
show that, by adopting a tensor network approach that reflects the geometry of
the lattice and is approximately contracted using belief propagation, we can
perform a classical simulation that is significantly more accurate and precise
than the results obtained from the quantum processor and many other classical
methods. We quantify the tree-like correlations of the wavefunction in order to
explain the accuracy of our belief propagation-based approach. We also show how
our method allows us to perform simulations of the system to long times in the
thermodynamic limit, corresponding to a quantum computer with an infinite
number of qubits. Our tensor network approach has broader applications for
simulating the dynamics of quantum systems with tree-like correlations.Comment: 18 Pages. 10 Figures. Updated to include improved BP-TNS data,
simulation of the infinite system and improved error quantificatio
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