Algorithms for entanglement renormalization

By Glen Evenbly, Guifré Vidal. 2007

289 citations recorded by the source corpus.

We describe an iterative method to optimize the multiscale entanglement renormalization ansatz for the low-energy subspace of local Hamiltonians on a D -dimensional lattice. For translation-invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of one-dimensional systems, namely, Ising, Potts, XX, and Heisenberg models. The potential to compute expected values of local observables, energy gaps, and correlators is investigated.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/15547df9e59a3c76abac720c0af0ef7ae0b30405

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “. In the PEPS and PEPO, the ranks are kept considerably smaller at a cost of employing 5th- or even 6th-order core tensors and the associated higher computational complexity with respect to the ranks [95,101]. Even with the PEPS/PEPO architectures, for very high-order tensors, the ranks (internal si”
Entanglement of purification: from spin chains to holography (2017, background): “[31] Other disentangling criteria are also possible - in this work we actually minimize the 2nd Renyi entropy for numerical efficiency (see Appendix C).”
Supervised Learning with Tensor Networks (2016, methodology): “While applications of MPS to machine learning have been a success, one aim of the present work is to have tensor networks play a more central role in developing learning models; another is to more easily incorporate powerful algorithms and tensor networks which generalize MPS developed by the physic”
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence (2015, methodology): “Vidal combined this idea with entanglement renormalization to formulate the Multiscale Entanglement Renormalization Ansatz (MERA) [16, 17], a family of tensor networks that efficiently approximate wave functions with long-range entanglement of the type exhibited by ground states of local scale-invar”
Hand-waving and interpretive dance: an introductory course on tensor networks (2016, background): “Despite this, physically relevant states often possess additional structure not found in arbitrary states, and as such do not exhibit this pathological complexity, allowing them to be efficiently described and studied.”
Efficient quantum state tomography. (2010, methodology): “Combinations of our techniques can be used to reconstruct other classes of states, such as tree tensor networks [36] and multiscale entanglement renormalization ansatz (MERA) states [37], for which efficient heuristics for minimizing local Hamiltonians are available.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “The Multiscale Entanglement Renormalization Ansatz (MERA) tensor network was first introduced by Vidal [2008], and for this network numerical algorithms to minimize some specific cost functions or local Hamiltonians used in quantum physics already exist [Evenbly and Vidal, 2009].”