GRADIENT FLOWS FOR OPTIMIZATION IN QUANTUM INFORMATION AND QUANTUM DYNAMICS: FOUNDATIONS AND APPLICATIONS

DOI10.1142/S0129055X10004053zbMath1208.49001arXiv0802.4195MaRDI QIDQ3585134

Steffen J. Glaser, Gunther Dirr, Uwe R. Helmke, Th. Schulte-Herbrüggen

Publication date: 19 August 2010

Published in: Reviews in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0802.4195

zbMATH Keywords

quantum controlRiemannian optimizationdouble-bracket flowstensor SVDconstrained optimization in quantum systemslow-rank approximation of tensorsRiemannian gradient flows and algorithms

Mathematics Subject Classification ID

Nonlinear programming (90C30) Numerical optimization and variational techniques (65K10) Variational problems in a geometric measure-theoretic setting (49Q20) Eigenvalues, singular values, and eigenvectors (15A18) Many-body theory; quantum Hall effect (81V70) Global Riemannian geometry, including pinching (53C20) Research exposition (monographs, survey articles) pertaining to differential geometry (53-02) Multilinear algebra, tensor calculus (15A69) Variational methods for eigenvalues of operators (49R05) Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control (49-02) Quantum control (81Q93)

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Uses Software

• L-BFGS

Cites Work

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• Dividing quantum channels
• Linear and nonlinear programming.
• Lie-semigroup structures for reachability and control of open quantum systems: Kossakowski-Lindblad generators form Lie wedge to Markovian channels
• Differential-geometric Newton method for the best rank-\((R _{1}, R _{2}, R _{3})\) approximation of tensors
• Minimizing a differentiable function over a differential manifold
• Finitely correlated states on quantum spin chains
• Abundance of translation invariant pure states on quantum spin chains
• Completely integrable gradient flows
• Curvatures of left invariant metrics on Lie groups
• Elementary inclusion relations for generalized numerical ranges
• On gradients of functions definable in o-minimal structures
• Representations of quasi-Newton matrices and their use in limited memory methods
• On the existence of homogeneous geodesics in homogeneous Riemannian manifolds
• Controllability of invariant systems on Lie groups and homogeneous spaces.
• Gradient flows computing the C-numerical range with applications in NMR spectroscopy
• The quantum query complexity of the hidden subgroup problem is polynomial
• Control systems on Lie groups
• Orthogonal Tensor Decompositions
• Rank-One Approximation to High Order Tensors
• Genuinely multipartite entangled states in higher dimensions: a generalization of balancedness
• Quantum Phase Transitions
• Quantum entanglement
• The density-matrix renormalization group
• Unifying Variational Methods for Simulating Quantum Many-Body Systems
• Area Laws in Quantum Systems: Mutual Information and Correlations
• Entangled Three-Qubit States without Concurrence and Three-Tangle
• An estimation of the controllability time for single-input systems on compact Lie Groups
• The Projected Gradient Method for Least Squares Matrix Approximations with Spectral Constraints
• Lie Theory for Quantum Control
• A Newton–Grassmann Method for Computing the Best Multilinear Rank-$(r_1,$ $r_2,$ $r_3)$ Approximation of a Tensor
• Quantum Measurement and Control
• Determination of the Transitivity of Bilinear Systems
• Quantum algorithms and the Fourier transform
• Quantum algorithms revisited
• C-numerical ranges andC-numerical radii
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• A Multilinear Singular Value Decomposition
• On the Best Rank-1 and Rank-(R₁ ,R₂ ,. . .,R_N) Approximation of Higher-Order Tensors
• Quantum error correction via codes over GF(4)
• Lie Theory and Control Systems Defined on Spheres
• Notions of controllability for bilinear multilevel quantum systems
• The significance of theC-numerical range and the localC-numerical range in quantum control and quantum information
• RelativeC-numerical ranges for applications in quantum control and quantum information
• System Theory on Group Manifolds and Coset Spaces
• Lie groups beyond an introduction
• Controllability of nonlinear systems