Tensor network representations from the geometry of entangled states

Tensor networks provide descriptions of strongly correlated quantum systems based on an underlying entanglement structure given by a graph of entangled states along the edges that identify the indices of the local tensors to be contracted. Considering a more general setting, where entangled states on edges are replaced by multipartite entangled states on faces, allows us to employ the geometric properties of multipartite entanglement in order to obtain representations in terms of superpositions of tensor network states with smaller effective dimension, leading to computational savings.

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On the geometry of tensor network states

Quantum Information and Computation ◽

10.26421/qic12.3-4-12 ◽

2012 ◽

Vol 12 (3&4) ◽

pp. 346-354

Author(s):

Joseph M. Landsburg ◽

Yang Qi ◽

Ke Ye

Keyword(s):

Graphical Models ◽

Matrix Multiplication ◽

Algebraic Statistics ◽

Geometric Complexity ◽

Network State ◽

Geometric Complexity Theory ◽

Tensor Networks ◽

Tensor Network ◽

Network States ◽

Tensor Network States

We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections of matrix multiplication (and re-expressions of matrix multiplication and its projections after changes of bases). Tensor Network States are also related to graphical models in algebraic statistics.

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A Geometric View on Quantum Tensor Networks

EPJ Web of Conferences ◽

10.1051/epjconf/202022602022 ◽

2020 ◽

Vol 226 ◽

pp. 02022

Author(s):

Alexander Tsirulev

Keyword(s):

Building Blocks ◽

Quantum Systems ◽

Mixed States ◽

Normal Coordinates ◽

Covariant Derivatives ◽

Tensor Networks ◽

Tensor Network ◽

Network States ◽

Tensor Network States ◽

Arbitrary Quantum

Tensor network states and algorithms play a key role in understanding the structure of complex quantum systems and their entanglement properties. This report is devoted to the problem of the construction of an arbitrary quantum state using the differential geometric scheme of covariant series in Riemann normal coordinates. The building blocks of the scheme are polynomials in the Pauli operators with the coefficients specified by the curvature, torsion, and their covariant derivatives on some base manifold. The problem of measuring the entanglement of multipartite mixed states is shortly discussed.

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Quantum Machine Learning Tensor Network States

Frontiers in Physics ◽

10.3389/fphy.2020.586374 ◽

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.

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Computable Rényi mutual information: Area laws and correlations

Quantum ◽

10.22331/q-2021-09-14-541 ◽

2021 ◽

Vol 5 ◽

pp. 541

Author(s):

Samuel O. Scalet ◽

Álvaro M. Alhambra ◽

Georgios Styliaris ◽

J. Ignacio Cirac

Keyword(s):

Mutual Information ◽

Correlation Functions ◽

Quantum Correlations ◽

Matrix Product ◽

Many Body ◽

Von Neumann ◽

Tensor Network ◽

Network States ◽

Tensor Network States ◽

Area Law

The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states like matrix product operators while preserving all desirable properties of a measure of correlations. In particular, we show that they obey a thermal area law in great generality, and that they upper bound all correlation functions. We also investigate their behavior on certain tensor network states and on classical thermal distributions.

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Plaquette renormalization scheme for tensor network states

Physical Review E ◽

10.1103/physreve.83.056703 ◽

2011 ◽

Vol 83 (5) ◽

Cited By ~ 14

Author(s):

Ling Wang ◽

Ying-Jer Kao ◽

Anders W. Sandvik

Keyword(s):

Renormalization Scheme ◽

Tensor Network ◽

Network States ◽

Tensor Network States

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Universal photonic quantum computation via time-delayed feedback

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1711003114 ◽

2017 ◽

Vol 114 (43) ◽

pp. 11362-11367 ◽

Cited By ~ 44

Author(s):

Hannes Pichler ◽

Soonwon Choi ◽

Peter Zoller ◽

Mikhail D. Lukin

Keyword(s):

Single Atom ◽

Many Body ◽

Cluster States ◽

Quantum Emitter ◽

Tensor Network ◽

Quantum Feedback ◽

Network States ◽

Tensor Network States ◽

Photonic Quantum Computation ◽

Time Delayed Feedback

We propose and analyze a deterministic protocol to generate two-dimensional photonic cluster states using a single quantum emitter via time-delayed quantum feedback. As a physical implementation, we consider a single atom or atom-like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits, while the mirror allows the implementation of feedback. We identify the class of many-body quantum states that can be produced using this approach and characterize them in terms of 2D tensor network states.

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