Multigrid Algorithms for Tensor Network States

Dolfi, Michele; Bauer, Bela; Troyer, Matthias; Ristivojević, Zoran

doi:10.1103/physrevlett.109.020604

Cited by 39 publications

(46 citation statements)

References 32 publications

(42 reference statements)

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“…This however, entails some technical difficulties and one can expect that using discrete-time MPEMs and a small time step should be the best option. Similarly, quantum many-body systems in continuous real-space have been treated efficiently using MPS with a sufficiently fine space discretization [62,63]. Other approximative schemes to simulate continuous-time stochastic dynamics are the dynamical replica analysis [64] that captures macroscopic observables and the cavity master equation method [65] that operates on local conditional probabilities.…”

Section: Models With Vertex-state Dependencementioning

confidence: 99%

The matrix product approximation for the dynamic cavity method

Barthel¹

2020

J. Stat. Mech.

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Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages -conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced in Phys. Rev. E 97, 010104(R) (2018), and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. Data for Glauber-Ising dynamics shows a linear growth of computation costs in time. In contrast to Monte Carlo simulations, the approach has a much better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. We discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.

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Section: Models With Vertex-state Dependencementioning

confidence: 99%

The matrix product approximation for the dynamic cavity method

Barthel¹

2020

J. Stat. Mech.

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“…In this way, computing Ψ|V int |Ψ with |Ψ represented by a matrix product state (MPS) 17 with bond dimension D scales as O(N t AD 3 ), that is, linearly with the system size. In combination with DMRG, this representation has been used to simulate interacting 1D models in the continuum limit [18][19][20][21] with controllable accuracy by arXiv:1907.06018v1 [cond-mat.str-el] 13 Jul 2019 systematically reducing the spacing l and increasing the bond dimension D.…”

Section: Introductionmentioning

confidence: 99%

Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

O’Rourke

Chan

2019

Phys. Rev. B

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In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions e −λr ij with varying exponents λ, each of which can be represented by a simple matrix product operator (MPO) with bond dimension D = 3. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states (MPS). However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to 2D and 3D by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters λ, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in 3D yields a D = 4 tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (r −1 ij ), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.

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“…where i, j label lattice sites, σ ∈ {α, β} labels spin, t is the kinetic energy matrix element, and a † , a, and n are fermion creation, annihilation, and number operators, respectively. As the spacing between grid points (h) goes to zero, the parameters scale as t ∝ h −2 and v ee ij ∝ h −1 ; these become exact representations of − 1 2 ∇ 2 and the continuum Coulomb potential 1/r ij with r ij |r i − r j | [26,28]. This simple form of the electronic structure Hamiltonian is especially suited to TNS algorithms as the Coulomb interaction is a pairwise operator as opposed to a general quartic operator when using a nonlocal basis, and Eq.…”

Section: Introductionmentioning

confidence: 99%

Efficient representation of long-range interactions in tensor network algorithms

O’Rourke

Chan

2018

Phys. Rev. B

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We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states. arXiv:1807.08378v2 [cond-mat.str-el]

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Multigrid Algorithms for Tensor Network States

Cited by 39 publications

References 32 publications

The matrix product approximation for the dynamic cavity method

The matrix product approximation for the dynamic cavity method

Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

Efficient representation of long-range interactions in tensor network algorithms

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