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Frank Verstraete - One of the best experts on this subject based on the ideXlab platform.
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tangent space methods for uniform Matrix Product states
arXiv: Strongly Correlated Electrons, 2019Co-Authors: Laurens Vanderstraeten, Jutho Haegeman, Frank VerstraeteAbstract:In these lecture notes we give a technical overview of tangent-space methods for Matrix Product states in the thermodynamic limit. We introduce the manifold of uniform Matrix Product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how Matrix Product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous Matrix Product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.
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Thermal states as convex combinations of Matrix Product states
Physical Review B, 2018Co-Authors: Mario Berta, Jutho Haegeman, Fernando G. S. L. Brandão, Volkher B. Scholz, Frank VerstraeteAbstract:We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of Matrix Product states. Apart from revealing new features of the entanglement structure of Gibbs states, our results provide a theoretical justification for the use of White's algorithm of minimally entangled typical thermal states. Furthermore, we shed new light on time dependent Matrix Product state algorithms which yield hydrodynamical descriptions of the underlying dynamics.
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variational optimization algorithms for uniform Matrix Product states
Physical Review B, 2018Co-Authors: Valentin Zaunerstauber, Laurens Vanderstraeten, Frank Verstraete, M T Fishman, Jutho HaegemanAbstract:We combine the density Matrix renormalization group (DMRG) with Matrix Product state tangent space concepts to construct a variational algorithm for finding ground states of one-dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform Matrix Product state algorithm (VUMPS) with infinite density Matrix renormalization group (IDMRG) and with infinite time evolving block decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long-range interactions and also for the simulation of two-dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation.
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Matrix Product unitaries structure symmetries and topological invariants
Journal of Statistical Mechanics: Theory and Experiment, 2017Co-Authors: Ignacio J Cirac, Frank Verstraete, Norbert Schuch, David PerezgarciaAbstract:Matrix Product vectors form the appropriate framework to study and classify one-dimensional quantum systems. In this work, we develop the structure theory of Matrix Product unitary operators (MPUs) which appear e.g. in the description of time evolutions of one-dimensional systems. We prove that all MPUs have a strict causal cone, making them quantum cellular automata (QCAs), and derive a canonical form for MPUs which relates different MPU representations of the same unitary through a local gauge. We use this canonical form to prove an index theorem for MPUs which gives the precise conditions under which two MPUs are adiabatically connected, providing an alternative derivation to that of (Gross et al 2012 Commun. Math. Phys. 310 419) for QCAs. We also discuss the effect of symmetries on the MPU classification. In particular, we characterize the tensors corresponding to MPU that are invariant under conjugation, time reversal, or transposition. In the first case, we give a full characterization of all equivalence classes. Finally, we give several examples of MPU possessing different symmetries.
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diagonalizing transfer matrices and Matrix Product operators a medley of exact and computational methods
Annual Review of Condensed Matter Physics, 2017Co-Authors: Jutho Haegeman, Frank VerstraeteAbstract:Transfer matrices and Matrix Product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and Matrix Product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, Matrix Product operator algebras, and numerical Matrix Product state methods for finding extremal eigenvectors of Matrix Product operators.
Jutho Haegeman - One of the best experts on this subject based on the ideXlab platform.
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tangent space methods for uniform Matrix Product states
arXiv: Strongly Correlated Electrons, 2019Co-Authors: Laurens Vanderstraeten, Jutho Haegeman, Frank VerstraeteAbstract:In these lecture notes we give a technical overview of tangent-space methods for Matrix Product states in the thermodynamic limit. We introduce the manifold of uniform Matrix Product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how Matrix Product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous Matrix Product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.
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Thermal states as convex combinations of Matrix Product states
Physical Review B, 2018Co-Authors: Mario Berta, Jutho Haegeman, Fernando G. S. L. Brandão, Volkher B. Scholz, Frank VerstraeteAbstract:We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of Matrix Product states. Apart from revealing new features of the entanglement structure of Gibbs states, our results provide a theoretical justification for the use of White's algorithm of minimally entangled typical thermal states. Furthermore, we shed new light on time dependent Matrix Product state algorithms which yield hydrodynamical descriptions of the underlying dynamics.
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variational optimization algorithms for uniform Matrix Product states
Physical Review B, 2018Co-Authors: Valentin Zaunerstauber, Laurens Vanderstraeten, Frank Verstraete, M T Fishman, Jutho HaegemanAbstract:We combine the density Matrix renormalization group (DMRG) with Matrix Product state tangent space concepts to construct a variational algorithm for finding ground states of one-dimensional quantum lattices in the thermodynamic limit. A careful comparison of this variational uniform Matrix Product state algorithm (VUMPS) with infinite density Matrix renormalization group (IDMRG) and with infinite time evolving block decimation (ITEBD) reveals substantial gains in convergence speed and precision. We also demonstrate that VUMPS works very efficiently for Hamiltonians with long-range interactions and also for the simulation of two-dimensional models on infinite cylinders. The new algorithm can be conveniently implemented as an extension of an already existing DMRG implementation.
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diagonalizing transfer matrices and Matrix Product operators a medley of exact and computational methods
Annual Review of Condensed Matter Physics, 2017Co-Authors: Jutho Haegeman, Frank VerstraeteAbstract:Transfer matrices and Matrix Product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and Matrix Product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, Matrix Product operator algebras, and numerical Matrix Product state methods for finding extremal eigenvectors of Matrix Product operators.
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anyons and Matrix Product operator algebras
Annals of Physics, 2017Co-Authors: Nick Bultinck, Jutho Haegeman, Frank Verstraete, Dominic J. Williamson, Mehmet B. Şahinoğlu, Michaël MariënAbstract:Abstract Quantum tensor network states and more particularly projected entangled-pair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of Matrix Product operators. In this paper, we present a systematic study of those Matrix Product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the Matrix Product operators we construct a C ∗ -algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S Matrix, fusion and braiding relations can readily be extracted from the idempotents. As the Matrix Product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.
Markus Reiher - One of the best experts on this subject based on the ideXlab platform.
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Spin-adapted Matrix Product states and operators
The Journal of chemical physics, 2016Co-Authors: Sebastian Keller, Markus ReiherAbstract:Matrix Product states (MPSs) and Matrix Product operators (MPOs) allow an alternative formulation of the density Matrix renormalization group algorithm introduced by White. Here, we describe how non-abelian spin symmetry can be exploited in MPSs and MPOs by virtue of the Wigner–Eckart theorem at the example of the spin-adapted quantum chemical Hamiltonian operator.
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an efficient Matrix Product operator representation of the quantum chemical hamiltonian
Journal of Chemical Physics, 2015Co-Authors: Sebastian Keller, Michele Dolfi, Matthias Troyer, Markus ReiherAbstract:We describe how to efficiently construct the quantum chemical Hamiltonian operator in Matrix Product form. We present its implementation as a density Matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of Matrix Product based DMRG. The latter variationally optimizes a class of ansatz states known as Matrix Product states, where operators are correspondingly represented as Matrix Product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a Matrix Product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries — abelian and non-abelian — and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
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an efficient Matrix Product operator representation of the quantum chemical hamiltonian
arXiv: Computational Physics, 2015Co-Authors: Sebastian Keller, Michele Dolfi, Matthias Troyer, Markus ReiherAbstract:We describe how to efficiently construct the quantum chemical Hamiltonian operator in Matrix Product form. We present its implementation as a density Matrix renormalization group (DMRG) algorithm for quantum chemical applications in a purely Matrix Product based framework. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from a viewpoint of Hilbert space decimation and attained a higher performance compared to straightforward implementations of Matrix Product based DMRG. The latter variationally optimizes a class of ansatz states known as Matrix Product states (MPS), where operators are correspondingly represented as Matrix Product operators (MPO). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a Matrix Product approach; for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
Dario Poletti - One of the best experts on this subject based on the ideXlab platform.
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Matrix Product states with adaptive global symmetries
Physical Review B, 2019Co-Authors: Chu Guo, Dario PolettiAbstract:Quantum many body physics simulations with Matrix Product States can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined before hand when constructing the tensors for the Matrix Product States algorithm. In this work, we present a Matrix Product States algorithm with an adaptive $U(1)$ symmetry. This algorithm can take into account of, or benefit from, $U(1)$ or $Z_2$ symmetries when they are present, or analyze the non-symmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.
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Matrix Product States with dynamically emerging global symmetries
arXiv: Quantum Physics, 2019Co-Authors: Chu Guo, Dario PolettiAbstract:Quantum many body physics simulations with Matrix Product States can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined before hand when constructing the tensors for the Matrix Product States algorithm. In this work, we present a Matrix Product States algorithm with a dynamical $U(1)$ symmetry. This algorithm can take into account of, or benefit from, $U(1)$ or $Z_2$ symmetries when they are present, or analyze the non-symmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.
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Matrix Product operators for sequence-to-sequence learning
Physical Review E, 2018Co-Authors: Chu Guo, Zhanming Jie, Dario PolettiAbstract:The method of choice to study one-dimensional strongly interacting many body quantum systems is based on Matrix Product states and operators. Such method allows to explore the most relevant, and numerically manageable, portion of an exponentially large space. Here we show how to construct a machine learning model in which Matrix Product operators are trained to implement sequence to sequence prediction, i.e. given a sequence at a time step, it allows one to predict the next sequence. We then apply our algorithm to cellular automata (for which we show exact analytical solutions in terms of Matrix Product operators), and to nonlinear coupled maps. We show advantages of the proposed algorithm when compared to conditional random fields and bidirectional long short-term memory neural network. To highlight the flexibility of the algorithm, we also show how it can perform classification tasks.
J I Cirac - One of the best experts on this subject based on the ideXlab platform.
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continuous Matrix Product states for quantum fields
Physical Review Letters, 2010Co-Authors: Frank Verstraete, J I CiracAbstract:We define Matrix Product states in the continuum limit, without any reference to an underlying lattice parameter. This allows us to extend the density Matrix renormalization group and variational Matrix Product state formalism to quantum field theories and continuum models in 1 spatial dimension. We illustrate our procedure with the Lieb-Liniger model.
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Matrix Product operator representations
New Journal of Physics, 2010Co-Authors: B Pirvu, Valentin Murg, J I Cirac, Frank VerstraeteAbstract:We show how to construct relevant families of Matrix Product operators (MPOs) in one and higher dimensions. These form the building blocks for the numerical simulation methods based on Matrix Product states and projected entangled pair states. In particular, we construct translationally invariant MPOs suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We show how these tools can be exploited for constructing new algorithms for simulating quantum spin systems.
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Matrix Product state representations
arXiv: Quantum Physics, 2006Co-Authors: David Perezgarcia, Frank Verstraete, Michael M. Wolf, J I CiracAbstract:This work gives a detailed investigation of Matrix Product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.
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Matrix Product states represent ground states faithfully
Physical Review B, 2006Co-Authors: Frank Verstraete, J I CiracAbstract:We quantify how well Matrix Product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.
