Unifying time evolution and optimization with matrix product states
By J. Haegeman, Christian Lubich, Ivan V. Oseledets, Bart Vandereycken, Frank Verstraete. 2014
724 citations recorded by the source corpus.
We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.
Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Condensed Matter Physics
View Paper: https://www.semanticscholar.org/paper/33fa5849d5aa262f805c506554e0db07ff383804
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In Atomic and Molecular Physics, and Optics
- L. Vanderstraeten
- Frank Verstraete
- J. Haegeman
- Frank Pollmann
- J. Hauschild
- A. Mirlin
- Xiaopeng Li
- Dong-Ling Deng
In Computational Theory and Mathematics
- T. Kohler
- Frank Verstraete
- J. Hauschild
- Sebastian Paeckel
- Andreas Swoboda
- Frank Pollmann
- U. Schollwock
- Salvatore R. Manmana
In Condensed Matter Physics
- L. Vanderstraeten
- Frank Verstraete
- Frank Pollmann
- J. Haegeman
- J. Hauschild
- Xiaopeng Li
- Dong-Ling Deng
- T. Kohler
Citations in the corpus
- Many-body localization and delocalization in large quantum chains (2018, methodology): “From the perspective of accuracy of MPS simulations, the worst-case scenario for a local Hamiltonian is realized when the von Neumann entanglement entropy for a bipartition of the chain into two parts grows linearly [39], which is known as volume-law entanglement and should be contrasted to the area”
- Tangent-space methods for uniform matrix product states (2018, methodology): “The MPS version of the TDVP [7, 15, 27] has been applied to spin chains [28], gauge theories [29, 30], and spin systems with long-range interactions [27, 31–34].”
- Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, background): “ks down". For example, TEBD does not even preserve the energy when the truncation is large. An improved algorithm based on the time dependent variational principle (TDVP) was introduced in Refs. [28,29] which performs a unitary evolution in the space of MPS with given bond dimension ˜ max. If w”
- Quantum Entanglement in Neural Network States (2017, methodology): “We remark that the DMRG/MPS-based simulations are particularly challenging for the above problem and would presumably require a substantially larger number of variational parameters than the RBM approach [9, 108].”
- Time-evolution methods for matrix-product states (2019, methodology): “[28, 29] was presented in Refs.”