From density-matrix renormalization group to matrix product states

By Ian P. McCulloch. 2007

335 citations recorded by the source corpus.

In this paper we give an introduction to the numerical density matrix renormalization group (DMRG) algorithm, from the perspective of the more general matrix product state (MPS) formulation. We cover in detail the differences between the original DMRG formulation and the MPS approach, demonstrating the additional flexibility that arises from constructing both the wavefunction and the Hamiltonian in MPS form. We also show how to make use of global symmetries, for both the Abelian and non-Abelian cases.

Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/49617779ea15e33eed9ab4ac2f4c2fd71f51d451

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In Atomic and Molecular Physics, and Optics

In Computational Theory and Mathematics

In Condensed Matter Physics

Citations in the corpus

The density matrix renormalization group for ab initio quantum chemistry (2014, background): “For the underlying MPS, this boils down to assuming that the rank-three MPS tensors are irreducible tensor operators of the symmetry group [163].”
OpenMolcas: From source code to insight. (2019, methodology): “The ﬂ exible nature of MPSs allows one to calculate excited states in an e ﬃ cient manner 110 and provides easy access to higher-order reduced density matrices and the evaluation of arbitrary correlation functions.”
Time-evolution methods for matrix-product states (2019, methodology): “Nowadays, DMRG methods formulated in the language of matrix-product states[9–14] (MPS) are the de facto standard for the investigation of one-dimensional systems.”
Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, neutral): “king tensor products of the operators), one can easily see that the product of the matrices does in fact yield the Hamiltonian (35). Further details of the MPO form of operators can be found in Refs. [11,62]. To derive the form of the matrices for a more complicated Hamiltonian, it can be useful to ”
Quantum optimization of maximum independent set using Rydberg atom arrays (2022, background): “The desired wavefunction can be represented as a matrix product state (MPS) [81, 82] of the form |Ψ〉 = ∑ σ1...σn ∑ l1...ln−1 T σ1l1 T σ2 l1l2 T σ3l2l3 · · · T σn ln−1 |σ1, . . . , σn〉, (S40) where T denote tensors with physical indices σ and link indices l. DMRG then provides an efficient method to ”
Hand-waving and interpretive dance: an introductory course on tensor networks (2016, background): “The above network is an example of what is referred to as matrix product operators (MPOs) [21, 27, 28].”
Quantum trajectories and open many-body quantum systems (2014, methodology): “This can be straight-forwardly generalised to higher-order Trotter decompositions [251], and a variety of other methods for efficiently propagating the state, especially Arnoldi methods [232, 252] are also available.”
The ITensor Software Library for Tensor Network Calculations (2020, background): “The two most well developed formats are the matrix product state (MPS) tensor network [4, 15, 16], also known as the tensor train [6], and the matrix product operator (MPO) tensor network [17,18].”