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 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.

scholarly journals Computable Rényi mutual information: Area laws and correlations

Quantum ◽  
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.

scholarly journals Universal photonic quantum computation via time-delayed feedback

2017 ◽  
Vol 114 (43) ◽  
pp. 11362-11367 ◽  
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.

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 On the geometry of tensor network states

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.

scholarly journals A Geometric View on Quantum Tensor Networks

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.

scholarly journals Tensor network representations from the geometry of entangled states

2020 ◽  
Vol 9 (3) ◽  
Author(s):  
Matthias Christandl ◽  
Angelo Lucia ◽  
Peter Vrana ◽  
Albert H. Werner
Keyword(s):  
General Setting ◽  
Entangled States ◽  
Strongly Correlated ◽  
Effective Dimension ◽  
Tensor Networks ◽  
Tensor Network ◽  
Correlated Quantum Systems ◽  
Network States ◽  
Tensor Network States ◽  
Entanglement Structure

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.

scholarly journals On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Mario Collura ◽  
Luca Dell'Anna ◽  
Timo Felser ◽  
Simone Montangero
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
State Of The Art ◽  
Detailed Comparison ◽  
Tensor Networks ◽  
The Neural Network ◽  
Tensor Network ◽  
Network Methods ◽  
Ferromagnetic Ising Model ◽  
Network States

In many cases, neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferromagnetic Ising model and the 2D antiferromagnetic Heisenberg model, addressing the lack of a detailed comparison of the expressiveness of these increasingly popular, variational ans"atze.

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