Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains

By Thomas K. Huckle, Konrad Waldherr, Thomas Schulte-Herbrüggen. 2013

16 citations recorded by the source corpus.

We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This article summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of MPS (Tensor Trains or Tensor Chains) and their canonical normal forms, we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context, we analyse different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This article provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.

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

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “There are variants of MPS/TT that reﬂect such symmetries in the low-rank decomposition, see [128,221,227] and the references therein.”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “7) (and was already shown in [24]) that a left normal representation satisfies the two gauge conditions”
Computations in quantum tensor networks (2012, background): “In [43] we present normal forms for mps which allow for expressing certain symmetry relations.”
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 ⊗”