Computations in quantum tensor networks

By Thomas K. Huckle, Thomas Schulte-Herbrüggen, Konrad Waldherr. 2012

71 citations recorded by the source corpus.

Abstract The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum many-body systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properties and efficient computations. The variational ansatz for this numerical approach leads to the minimization of the Rayleigh quotient. The Alternating Least Squares technique is then applied to break down the eigenvector computation to problems of appropriate size, which can be solved by classical methods. Efficient computations require fast computation of the matrix-vector product and of the inner product of two decomposed vectors. To this end, both appropriate representations of vectors and efficient contraction schemes are needed. Here approaches from many-body quantum physics for one-dimensional and two-dimensional systems (Matrix Product States and Projected Entangled Pair States) are treated mathematically in terms of tensors. We give the definition of these concepts, bring some results concerning uniqueness and numerical stability and show how computations can be executed efficiently within these concepts. Based on this overview we present some modifications and generalizations of these concepts and show that they still allow efficient computations such as applicable contraction schemes. In this context we consider the minimization of the Rayleigh quotient in terms of the parafac (CP) formalism, where we also allow different tensor partitions. This approach makes use of efficient contraction schemes for the calculation of inner products in a way that can easily be extended to the mps format but also to higher dimensional problems.

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

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Computational Mathematics

In Condensed Matter Physics

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “The connection between TT and MPS has been explained in [127].”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, background): “A number of recent, more mathematically oriented papers illustrate how these ideas can be adapted and transferred to other applications [3, 10, 12, 13, 14, 27].”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, background): “The matrix product type tensor approximation was proved to be efficient in high-dimensional electronic/molecular structure calculations, in quantum computing and in stochastic PDEs (see the survey publications [27] [42] [20]).”
Tensor Ring Decomposition (2016, background): “Recently, tensor networks, considered as the generalization of tensor decompositions, have emerged as the potentially powerful tools for analysis of large-scale tensor data [23], [24], [25], [26].”
Changing the topology of Tensor Networks (2012, background): “The Motivation for this conversion is due to the fact that the complexity of contracting a PEPS tensor is very high and the optimization procedure is not stable (see [13] for an approximated contraction scheme).”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “Also, in quantum physics the ALS algorithm is called one-site DMRG, while the Modified ALS is known as the two-site DMRG (for more detail, see Part 2 and [128, 201, 215, 232, 267, 271, 280] and references therein”
Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains (2013, background): “[11]) to consider two neighbouring pairs (cf.”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “For example, the Heisenberg (XYZ) model with open boundary conditions [51, 52], acting on d ⊗”
Algorithm 941 (2014, methodology): “1 Tensor Network Diagrams In the following, we briefly introduce tensor network diagrams (also called Penrose diagrams) to conveniently describe algorithms for contractions of tensors in HTD, see also [16, 17].”
The geometry of algorithms using hierarchical tensors (2013, background): “The rationale for treating rank-structured tensors as a manifold Besides their longstanding tradition in the computational physics community (see [20, 23] for recent overviews), the (Q)TT and HT formats have more recently been used, amongst others, in [24–31] for solving a variety of highdimensional”